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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors consider \textit{proper permutations} \(w \in S_n\), i.e., ones which satisfy \(\ell(w) - \binom{d(w) + 1}{2} \leq n\), where \(\ell(w)\) is the number of inversions and \(d(w)\) is the number of left descents of~\(w\). One of the main results of the article is that the probability that a random permutation \(w \in S_n\) is proper goes to zero in the limit when \(n \rightarrow \infty\). A very important aspect of this result is its relation to geometry: properness of \(w\) is related to the Schubert variety \(X_w\) being spherical. We say that \(X_w\) is spherical if it has a dense orbit of a Borel subgroup of some \(L_I\), a group of invertible block diagonal matrices, where blocks are determined by a set \(I\) of left descents of~\(w\). The authors conclude that the probability that for a random permutation \(w \in S_n\) the Schubert variety \(X_w\) is spherical goes to zero in the limit when \(n \rightarrow \infty\). Finally, the authors consider the notion of \(w \in S_n\) being \(I\)-spherical, introduced by \textit{R. Hodges} and \textit{A. Yong} in [J. Lie Theory, 32(2), 447--474 (2022; Zbl 1486.14070)], and show that the probability of \(w\) being \(I\)-spherical goes to zero in the limit when \(n \rightarrow \infty\). This result settles a conjecture from the article cited above. Schubert varieties; spherical varieties; proper permutations Grassmannians, Schubert varieties, flag manifolds, Probabilistic methods in group theory Proper permutations, Schubert geometry, and randomness

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The purpose of this article is to determine the Hilbert function of the tangent cone to points in a Schubert subvariety of a Grassmannian. \textit{S. S. Abhyankar} [``Enumerative combinatorics of Young tableaux'', Pure and Applied Mathematics 115 (1988; Zbl 0643.05001)] and \textit{J. Herzog} and \textit{N. V. Trung} [Adv. Math. 96, No. 1, 1--37 (1992; Zbl 0778.13022)] gave formulas for the multiplicity and the Hilbert function of the quotient ring of the determinantal ideal of the tangent cone at the identity coset, in the ring of the tangent space of the Grassmann variety. \textit{V. Lakshmibai} and \textit{J. Weyman} [Adv. Math. 84, No. 2, 179--208 (1990; Zbl 0729.14037)] gave a recursive formula for the multiplicity at any point, and from this formula \textit{J. Rosenthal} and \textit{A. Zelevinsky} [J. Algebr. Comb. 13, No. 2, 213--218 (2001; Zbl 1015.14025)] obtained a closed form for the multiplicity at any point. \textit{V. Kreiman} and \textit{V. Lakshmibai} [in: Algebra and algebraic geometry with applications, 553--563 (2002; Zbl 1092.14060)] obtained an expression for the Hilbert function at the identity coset in terms of the combinatorics of the Weyl group, and they recovered the interpretation of the multiplicity due to Herzog and Trung. They also reformulated their main result in terms of generic determinantal minors. In addition they conjectured an expression for the Hilbert function and the multiplicity when \(x\) is any point. In the present article the approach of Kreiman and Lakshmibai is clarified, and their expression for the Hilbert function and the multiplicity is extended to all points, as is their combinatorial interpretation of the multiplicity. \textit{C. Krattenthaler} [Sémin. Lothar. Comb. 45, B45c, 11 p. (2000; Zbl 0965.14023)] has given an interpretation of the Rosenthal-Zelevinsky formula using combinatorial techniques, and he proves the multiplicity formula of Kreiman and Lakshmibai and shows that their conjecture about the Hilbert function is equivalent to a certain finite problem. The approach of Krattenthaler is completely different from that of the present article. While Krattenthaler explores the combinatorial interpretations of the formula, the present article, in the spirit of Lakshmibai-Weyman and Kreiman-Lakshmibai, uses standard monomial theory to translate the problems from geometry to combinatorics. tangent cone; determinantal ideals; standard monomial theory; multiplicity V. Kodiyalam, K. Raghavan, Hilbert functions of points on Schubert varieties in the Grassmannian, J. Algebra 270 (2003), 28--54. Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert functions of points on Schubert varieties in Grassmannians.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0695.00010.] The purpose of the paper under review is to give a proof of six formulas by Schubert (two of which he proved and four of which he only conjectured) concerning the number of double contacts among the curves of two families of plane curves. The method consists in finding bases of the Chow groups of the Hilbert scheme of length 2 subschemes of the point- line incidence variety. This approach turns out to be much simpler than the one using the space of triangles as suggested by Schubert. As a byproduct, the authors obtain proofs of the classical formulas on triple contacts (i.e., single contacts of third order) between two such families of curves. flag variety; number of double contacts; families of plane curves; Chow groups; triple contacts Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Proof of Schubert's conjectures on double contacts

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. Using a criterion of Vakil and a special position argument due to Schubert, this follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, an easy combinatorial injection proves the inequality. For the remaining cases, we use that these Kostka numbers appear in tensor product decompositions of \(\mathfrak{sl}_2\mathbb{C}\)-modules. Interpreting the tensor product as the action of certain commuting Toeplitz matrices and using a spectral analysis and Fourier series rewrites the inequality as the positivity of an integral. We establish the inequality by estimating this integral. Kostka numbers; Galois groups; Schubert calculus; Schubert varieties Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations An inequality of Kostka numbers and Galois groups of Schubert problems

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations \(w\) with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\) for some \(h\). Kazhdan-Lusztig polynomials; Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Hecke algebras and their representations Permutations with Kazhdan-Lusztig polynomial \(P_{id,w}(q)=1+q^{h}\)

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We apply the previous calculations of Chow-Witt rings of Grassmannians to develop an oriented analogue of the classical Schubert calculus. As a result, we get complete diagrammatic descriptions of the ring structure in Chow-Witt rings and twisted Witt groups. In the resulting arithmetic refinements of Schubert calculus, the multiplicity of a solution subspace is a quadratic form encoding additional orientation information. We also discuss a couple of applications, such as a Chow-Witt version of the signed count of balanced subspaces of \textit{L. M. Fehér} and \textit{K. Matszangosz} [Period. Math. Hung. 73, No. 2, 137--156 (2016; Zbl 1389.14026)]. Chow-Witt rings; characteristic classes; Grassmannians Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Motivic cohomology; motivic homotopy theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Oriented Schubert calculus in Chow-Witt rings of Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article we explain how the coordinate ring of each (open) Schubert variety in the Grassmannian can be identified with a cluster algebra, whose combinatorial structure is encoded using (target labelings of) Postnikov's plabic graphs. This result generalizes a theorem of \textit{J. S. Scott} [Proc. Lond. Math. Soc. (3) 92, No. 2, 345--380 (2006; Zbl 1088.22009)] for the Grassmannian, and proves a folklore conjecture for Schubert varieties that has been believed by experts since Scott's result [loc. cit.], though the statement was not formally written down until \textit{G. Muller} and \textit{D. E. Speyer} [ibid. (3) 115, No. 5, 1014--1071 (2017; Zbl 1408.14154)] explicitly conjectured it. To prove this conjecture, we use a result of \textit{B. Leclerc} [Adv. Math. 300, 190--228 (2016; Zbl 1375.13036)] who used the module category of the preprojective algebra to prove that coordinate rings of many Richardson varieties in the complete flag variety can be identified with cluster algebras. Our proof also uses a construction of \textit{R. Karpman} [J. Comb. Theory, Ser. A 142, 113--146 (2016; Zbl 1337.05114)] to build plabic graphs associated to reduced expressions. We additionally generalize our result to the setting of skew-Schubert varieties; the latter result uses generalized plabic graphs, that is, plabic graphs whose boundary vertices need not be labeled in cyclic order. plabic graphs; module category of the preprojective algebra Combinatorial aspects of groups and algebras, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Cluster algebras Cluster structures in Schubert varieties in the Grassmannian

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For each infinite series of the classical Lie groups of type \(B\), \(C\) or \(D\), we construct a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the appropriate flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. They are also positive in a certain sense, and when indexed by maximal Grassmannian elements, or by the longest element in a finite Weyl group, these polynomials can be expressed in terms of the factorial analogues of Schur's \(Q\)- or \(P\)-functions defined earlier by Ivanov. double Schubert polynomials; equivariant cohomology Ikeda, T.; Mihalcea, L.; Naruse, H., \textit{double Schubert polynomials for the classical groups}, Adv. Math., 226, 840-886, (2011) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Double Schubert polynomials for the classical groups

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to \textit{N. Bergeron} and \textit{F. Sottile} [Duke Math. J. 95, 373--423 (1998; Zbl 0939.05084)] in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs. Schubert polynomial; Bruhat order; Littlewood-Richardson coefficient Lenart, C., Sottile, F.: Skew Schubert polynomials. Proceedings of the American Mathematical Society 131(11), 3319--3328 (2003) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Combinatorics of partially ordered sets Skew Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials One way to reformulate the celebrated theorem of Beilinson is that \((\mathcal{O}(-n),\dots , \mathcal{O})\) and \((\Omega^n(n), \dots , \Omega^1 (1), \mathcal{O})\) are strong complete exceptional sequences in \(D^b(\text{Coh}\,\mathbb P^n)\), the bounded derived category of coherent sheaves on \(\mathbb P^n\). In a series of papers, M. M. Kapranov generalized this result to flag manifolds of type \(A_n\) and quadrics. In another direction, Y. Kawamata has recently proven existence of complete exceptional sequences on toric varieties. Starting point of the present work is a conjecture of F. Catanese which says that on every rational homogeneous manifold \(X=G/P\), where \(G\) is a connected complex semisimple Lie group and \(P\subset G\) a parabolic subgroup, there should exist a complete strong exceptional poset and a bijection of the elements of the poset with the Schubert varieties in \(X\) such that the partial order on the poset is the order induced by the Bruhat-Chevalley order. An answer to this question would also be of interest with regard to a conjecture of \textit{B. Dubrovin} [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 315--326 (1998; Zbl 0916.32018)] which has its source in considerations concerning a hypothetical mirror partner of a projective variety \(Y\): There is a complete exceptional sequence in \(D^b(\text{Coh}\, Y)\) if and only if the quantum cohomology of \(Y\) is generically semisimple (the complete form of the conjecture also makes a prediction about the Gram matrix of such a collection). A proof of this conjecture would also support M. Kontsevich's homological mirror conjecture, one of the most important open problems in applications of complex geometry to physics today. The goal of this work will be to provide further evidence for F. Catanese's conjecture, to clarify some aspects of it and to supply new techniques. In section 2 it is shown among other things that the length of every complete exceptional sequence on \(X\) must be the number of Schubert varieties in \(X\) and that one can find a complete exceptional sequence on the product of two varieties once one knows such sequences on the single factors, both of which follow from known methods developed by Rudakov, Gorodentsev, Bondal et al. Thus one reduces the problem to the case \(X=G/P\) with \(G\) simple. Furthermore it is shown that the conjecture holds true for the sequences given by Kapranov for Grassmannians and quadrics. One computes the matrix of the bilinear form on the Grothendieck \(K\)-group \(K_{\circ}(X)\) given by the Euler characteristic with respect to the basis formed by the classes of structure sheaves of Schubert varieties in \(X\); this matrix is conjugate to the Gram matrix of a complete exceptional sequence. Section 3 contains a proof of theorem 3.2.7 which gives complete exceptional sequences on quadric bundles over base manifolds on which such sequences are known. This enlarges substantially the class of varieties (in particular rational homogeneous manifolds) on which those sequences are known to exist. In the remainder of section 3 we consider varieties of isotropic flags in a symplectic resp. orthogonal vector space. By a theorem due to Orlov one reduces the problem of finding complete exceptional sequences on them to the case of isotropic Grassmannians. For these, theorem 3.3.3 gives generators of the derived category which are homogeneous vector bundles; in special cases those can be used to construct complete exceptional collections. In subsection 3.4 it is shown how one can extend the preceding method to the orthogonal case with the help of theorem 3.2.7. In particular we prove theorem 3.4.1 which gives a generating set for the derived category of coherent sheaves on the Grassmannian of isotropic 3-planes in a 7-dimensional orthogonal vector space. Section 4 is dedicated to providing the geometric motivation of Catanese's conjecture and it contains an alternative approach to the construction of complete exceptional sequences on rational homogeneous manifolds which is based on a theorem of M. Brion and cellular resolutions of monomial ideals á la Bayer/Sturmfels. We give a new proof of the theorem of Beilinson on \(\mathbb P^n\) in order to show that this approach might work in general. We also prove theorem 4.2.5 which gives a concrete description of certain functors that have to be investigated in this approach. Christian Böhning, Derived categories of coherent sheaves on rational homogeneous manifolds, Doc. Math. 11 (2006), 261 -- 331. Grassmannians, Schubert varieties, flag manifolds, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories Derived categories of coherent sheaves on rational homogeneous manifolds

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0716.00007.] The author studies the Chern characters of a hypersurface with singularities, as defined by Wu. To define these characters, he introduces a new family of symmetric functions in two sets of variables \(\{x_ i\}\), \(\{y_ i\}\). For any family \(\{x_ 1,...,x_ n\}\), \(\{y_ 1,...,y_ m\}\) and for any integer \(i=1,...,n\), the symmetric polynomial \(CH_ i(x,y)\) is defined by \(CH_ i(x,y)=\sum^{i}_{j=0}(-1)^ j\left( \begin{matrix} n+1-j\\ i-j\end{matrix} \right)e^{i-j}(y)e^ i(x) \), where \(e(x)=x_ 1+...+x_ n\) and \(e(y)=y_ 1+...+y_ m\). - For any integer \(k\leq n\), and for any partition \(\pi =(\pi_ 1,...,\pi_ j)\) of k, i.e. \(\pi_ 1\geq \pi_ 2\geq...\geq \pi_ j\) and \(\pi_ 1+...+\pi_ j=k\), the symmetric polynomial \(CH_{\pi}(x,y)\) corresponding to the partition \(\pi\) is defined as the product \(CH_{\pi}(x,y)=CH_{\pi_ 1}(x,y)\cdot CH_{\pi_ 2}(x,y)...CH_{\pi_ j}(x,y)\). Every polynomial \(CH_{\pi}(x,y)\) can be expressed as a sum \(\sum^{k}_{i=0}a_ i(\pi)e^{k-i}(y)e^ i(x).\) Then the author shows that the dimension of the linear space \(CH^{(k)}(x,y)\) generated by te symmetric polynomials \(CH_{\pi}(x,y)\), with \(\pi\) a partition of k, is equal to k, and he gives a base \((CH_{\eta_ 1}(x,y),...,CH_{\eta_ k}(x,y))\), where the polynomial \(CH_{\eta_ i}(x,y)\) is associated with the ``hooked'' partition \(\eta_ i=(i,1,...,1)\). He defines a notion of positivity of the elements of the subspace \({\mathbb{E}}^{(k)}(x)\) of polynomials of \(CH^{(k)}(x,y)\) which are independent of the variables \(y_ 1,...,y_ m\). Every element W(x) of \({\mathbb{E}}^{(k)}(x)\) is equal to \(C.e^ k(x)\), with C a constant, and W(x) is positive if \(C>0\). If for any \(k\geq 2\), we set \(c_ 1(k)=(-1)^{k-1}\frac{n}{k}\left( \begin{matrix} n-1\\ k-2\end{matrix} \right)\) and \(c_ i(k)=(-1)^{k-i}\left( \begin{matrix} n+1-i\\ k-i\end{matrix} \right)\), \(i=2,...,k\), then the element \(W_ k(x)=\sum^{k}_{i=1}c_ i(k)\cdot CH_{\eta_ 1}(x,y) \) of \(CH^{(k)}(x,y)\) belongs to \({\mathbb{E}}^{(k)}\) and \((-1)^ kW_ k(x)\) is positive. - When \(k\geq 4\), the number of partitions of k is greater than k, then there exist some relations between the symmetric polynomials \(CH_{\pi}(x,y)\). The author gives some of them, for instance: \(C_ k\cdot W_{\pi_ 1}(x)\cdot W_{\pi_ 2}(x)...W_{\pi_ j}(x)=C_{\pi_ 1}\cdot C_{\pi_ 2}...C_{\pi_ j}\cdot W_ k(x)\), where \(\pi =(\pi_ 1,...,\pi_ j)\) is a partition of k with \(\pi_ j\geq 2\), and where \(C_ i=c_ 1(i)+...+c_ i(i).\) According to the definition of Wu, for any hypersurface \(V_ n\) with singularities in an \((n+1)\)-dimensional smooth complex projective space, the author defines the Chern classes \(CH_ i(V_ n)\), \(i=1,...,n\), by \(CH_ i(V_ n)=\sum^{i}_{j=0}(-1)^ j\left( \begin{matrix} n+1-j\\ i- j\end{matrix} \right)(p^{i-j}q^ j) \), where \(p=p(V_ n)\) and \(q=q(V_ n)\) are the Ehresmann classes of \(V_ n\). Then for any partition \(\pi\) of an integer k, one can define the Chern class \(CH_{\pi}(V_ n)\); this class belongs to the group of (n-k)-dimensional algebraic equivalence classes \(A_{n-k}(V_ n)\). Then the author can use the results on the symmetric polynomials \(CH_{\pi}(x,y)\) to find some positivity properties and some relations for the Chern classes \(CH_{\pi}(V_ n)\) of \(V_ n\). positivity of polynomials; Chern characters of a hypersurface with singularities; Chern classes; Ehresmann classes Singularities of surfaces or higher-dimensional varieties, Characteristic classes and numbers in differential topology Symmetric functions and the Chern characters of a hypersurface with singularities

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce two families of symmetric functions with an extra parameter \(t\) that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when \(t=1\). The families are defined by a statistic on combinatorial objects associated to the type-\(A\) affine Weyl group and their transition matrix with Hall-Littlewood polynomials is \(t\)-positive. We conjecture that one family is the set of \(k\)-atoms. \(k\)-Schur functions; Pieri rule; Bruhat order; Hall-Littlewood polynomials Dalal, A.; Morse, J., ABC's of the affine Grassmannian, (DMTCS Proceedings, (2013)) Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus A \(t\)-generalization for Schubert representatives of the affine Grassmannian

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author asks for the connection between a recent generalization of the ``second fundamental theorem'' of invariant theory due to Abhyankar and the geometry of the flag variety \(FL(n)=\sqcup W(\tau) \), where the W(\(\tau)\) are the Bruhat cells of FL(n). The Zariski closure of a W(\(\tau)\) in FL(n) is called a Schubert variety X(\(\tau)\) in FL(n). The fundamental theorem claims the primality of the determinantal ideals \(I_ p(X)\subset k[X]\) where \(X=(X_{ij})\) is a matrix of indeterminates over a field k. In the generalization mentioned above one investigates ideals \(I_ p({\mathcal L})\), where \({\mathcal L}\) is a certain subset (ladder) of X. The corresponding affine variety is denoted by V(p,\(\mathcal L)\). In case \(\mathcal L\) is a rectangular ladder, \(V(p,\mathcal L)\) is a determinantal locus and therefore the corresponding X(\(\tau)\) is called a determinantal-type Schubert variety. In section 4 (see theorems 3, 5, 6) it is shown that in general \(V(p,\mathcal L)\) is an intersection of determinantal loci and consequently each Schubert variety \(X(\tau)\) in FL(n) is an intersection of determinantal-type Schubert varieties. There are \(\binom{n+1}3\) determinantal-type Schubert varieties of FL(n) of which \((n-1)\) are divisors in FL(n), and a determinantal-type Schubert variety has codimension \(\leq n^ 2/4\) in FL(n). These intersections are ideal-theoretic and the proofs are combinatorial. Therefore the main section of this paper is section 2 on bivectors and permutations which makes up almost all the paper. combinatorial proofs; second fundamental theorem of invariant theory; flag variety; Schubert variety; determinantal ideals Mulay, S. B.: Determinantal loci and the flag variety. Adv. math. 74, 1-30 (1989) Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties Determinantal loci and the flag variety

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials {Let \(Fl_n\) be the manifold of complete flags in the \(n\)-dimensional vector space \(\mathbb C^n\). Inspired from ideas from string theory, recently the concept of quantum cohomology ring \(QH^*(X,\mathbb Z)\) of a Kähler algebraic manifold \(X\) has been defined. Then \[ QH^*(Fl_n,\mathbb Z)\cong H^*(Fl_n,\mathbb Z) \otimes\mathbb Z[q_1,\dots,q_{n-1}], \] where \(H^*(X,\mathbb Z)\) is the usual cohomology ring of \(Fl_n\) and \(q_1,\dots,q_{n-1}\) are formal variables (deformation parameters). So, the additive structures of the two cohomology rings are essentially the same. The multiplicative structure of \(H^*(X,\mathbb Z)\) can be recuperated from the multiplicative structure of \(QH^*(X,\mathbb Z)\) by taking \(q_1=\cdots=q_{n-1}=0\). The structure constants for the quantum cohomology are the 3-point Gromov-Witten invariants of genus zero. Recently, Givental, Kim and Ciocan-Lafontaine found a canonical isomorphism \[ QH^*(X,\mathbb Z)\cong\mathbb Z[q_1,\dots,q_{n-1}][x_1,\dots,x_n]/I_n^q, \] where \(x_1,\dots,x_n\) are variables and \(I_n^q\) is a certain ideal which can be explicitly described. This isomorphism extends an old isomorphism of Borel for the ordinary cohomology ring. The next problem naturally arising in the theory of quantum cohomology of the flag manifolds is to find an algebraic/combinatorial method for computing the structure constants of quantum multiplication in the basis of Schubert classes (the Gromov-Witten invariants). The aim of the paper under review is to solve this problem completely.} flag varieties; Schubert varieties; quantum cohomology ring; complete flags; Gromov-Witten invariants S. Fomin, S. Gelfand, and A. Postnikov, Quantum Schubert polynomials , J. Amer. Math. Soc., 168 (1997), 565--596. JSTOR: Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Quantum Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of the paper under review is to give a new proof of a recent result of [\textit{W. Fulton} and \textit{C. Woodward} [J. Algebr. Geom. 13, 641--661 (2004; Zbl 1081.14076)] related to the smallest degree that appears in the expansion of the product of two Schubert cycles in the small quantum cohomology ring of a Grassmann variety. The author's approach is combinatorial, and this method also yields an alternative characterization of this smallest degree in terms of the rim hook formula for the quantum product. quantum cohomology ring; Grassmann varieties; Schubert cycles; Gromov-Witten invariants A. Yong. ''Degree bounds in quantum Schubert calculus''. Proc. Amer. Math. Soc. 131 (2003), pp. 2649--2655.DOI. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Degree bounds in quantum Schubert calculus

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study interpolation by Grassmannian Schubert polynomials (Schur functions). We prove versions of the Sturmfels-Zelevinsky formula for the product of the maximal minors of rectangular matrices corresponding to elementary symmetric functions and Schur functions, and deduce from them generalizations of formulae for the Cauchy-Vandermonde determinant and Cauchy's formula for Schur functions. We define generalizations of higher Bruhat orders whose elements encode connected components of configuration spaces, and also generalizations of discriminantal Manin-Schechtman arrangements. Schur functions; symmetric functions; Cauchy-Vandermonde determinant; Cauchy's formula; Manin-Schechtman arrangements. Symmetric functions and generalizations, Configurations and arrangements of linear subspaces, Determinants, permanents, traces, other special matrix functions, Numerical interpolation Symmetric function interpolation and alternating higher Bruhat orders.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors compute the Witt groups of split Grassmann varieties, over any regular base \(X\). They prove that the total Witt group of the Grassmannian is a free module over the total Witt ring of \(X\). Remark that also the Chow group, or the Grothendieck group, are free over \(X\) with a basis indexed by all Young diagrams. The authors provide an explicit basis of the total Witt group indexed by a class of Young diagrams which they call even Young diagrams. Recall that the total Witt group is a sum of all the Witt groups depending on a shift \(i\in \mathbb{Z}/4\) and a twist \(L\in \mathrm{Pic} (X)/2\). The cited basis consists of homogeneous elements; moreover the shift and the twist can be read on the corresponding Young diagram. In particular, this fact allows the author to describe the unshifted and untwisted Witt group. The elements of the basis of the total Witt group are defined as push-forwards of the unit form of certain desingularized Schubert varieties. Remark that pushing the unit form is not always possible. The condition for a Young diagram to be even implies the existence of such a push-forward, but it is not necessary. The computation of Witt groups of a Grassmann variety is harder than the computation of cohomology groups or Chow groups because of the following fact. The classical computation proceeds by induction, using the closed embedding of a smaller Grassmannian \(\mathrm{Gr}_X(d,n-1)\) inside \(\mathrm{Gr}_X(d,n)\), whose open complement \(U\) is an affine bundle over another smaller Grassmannian \(\mathrm{Gr}_X(d-1,n)\). Moreover the restriction morphism from the big Grassmannian \(\mathrm{Gr}_X(d,n)\) to the open \(U\) is split surjective. For Witt groups the restriction morphism is not even surjective. In other words, the connecting homomorphism in the localization long exact sequence in not zero in general. Grassmann variety; Witt group; triangulated category; cellular decomposition Paul Balmer and Baptiste Calmès, Witt groups of Grassmann varieties, J. Algebraic Geom. 21 (2012), no. 4, 601 -- 642. Grassmannians, Schubert varieties, flag manifolds, Algebraic theory of quadratic forms; Witt groups and rings, Derived categories, triangulated categories Witt groups of Grassmann varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X_w^v\) be a Richardson variety in the full flag variety \(X\) associated to a symmetrizable Kac-Moody group \(G\). Recall that \(X_w^v\) is the intersection of the finite-dimensional Schubert variety \(X_w\) with the finite-codimensional opposite Schubert variety \(X^v\). We give an explicit \(\mathbb{Q}\)-divisor \(\Delta\) on \(X_w^v\) and prove that the pair \((X_w^v, \varDelta)\) has Kawamata log terminal singularities. In fact, \(-K_{X_w^v}-\varDelta\) is ample, which additionally proves that \(X_w^v,\varDelta\) is log Fano. We first give a proof of our result in the finite case (i.e., in the case when \(G\) is a finite-dimensional semisimple group) by a careful analysis of an explicit resolution of singularities of \(X_w^v\) (similar to the Bott-Samelson-Demazure-Hansen resolution of the Schubert varieties). In the general Kac-Moody case, in the absence of an explicit resolution of \(X_w^v\) as above, we give a proof that relies on the Frobenius splitting methods. In particular, we use Mathieu's result asserting that the Richardson varieties are Frobenius split, and combine it with a result by \textit{N. Hara} and \textit{K.-i. Watanabe} [J. Algebr. Geom. 11, No. 2, 363--392 (2002; Zbl 1013.13004)] relating Frobenius splittings with log canonical singularities. Grassmannians, Schubert varieties, flag manifolds, Minimal model program (Mori theory, extremal rays) Richardson varieties have Kawamata log terminal singularities

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A fundamental open problem in combinatorial representation theory and algebraic geometry is to give closed formulas for calculations in the cohomology ring of the full flag variety \(\mathcal F:=SL_n/B\) in terms of the basis of Schubert classes. The paper under review answers a part of this question by expressing the classes of certain subvarieties of the flag variety, called \textit{regular semisimple Hessenberg varieties}, in terms of the Schubert classes. Regular semisimple Hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory, numerical analysis, representation theory, algebraic geometry, and combinatorics. The authors give a ``Giambelli formula'' expressing the classes of regular semisimple Hessenberg varieties in terms of Chern classes. In fact, they show that the cohomology class of each regular semisimple Hessenberg variety is the specialization of a certain double Schubert polynomial, giving a natural geometric interpretation to such specializations. They also decompose such classes in terms of the Schubert basis for the cohomology ring of the flag variety. The coefficients obtained are nonnegative, and they give closed combinatorial formulas for the coefficients in many cases. They introduce a closely related family of schemes called regular nilpotent Hessenberg schemes, and use their results to determine when such schemes are reduced. Schubert polynomials; Hessenberg varieties Anderson, D; Tymoczko, J, \textit{Schubert polynomials and classes of Hessenberg varieties}, J. Algebra, 323, 2605-2623, (2010) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Schubert polynomials and classes of Hessenberg varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors prove a Littlewood-Richardson rule for the \(K\)-theory of odd orthogonal Grassmannians. That is, consider the variety \(X\) of isotropic \(n\)-planes in \(C^{2n+1}\) (with respect to a non-degenerate symmetric bilinear form). In the \(K\)-theory ring of \(X\) a natural basis is given by the classes of the structure sheaves of geometrically defined subvarieties, the Schubert varieties. Schubert varieties are parametrized by combinatorial objects, called shifted Young diagrams. Multiplication in \(K\)-theory hence defines structure constants depending on three shifted Young diagrams. One would like to find a combinatorial rule (``Littlewood-Richardson rule'') calculating these structure constants from the three Yong diagrams. Based on earlier results (and conjectures) of Thomas-Yong, and a recent result by Buch-Ravikumar, the authors prove such a combinatorial rule. Technically, Theorem 1.1 proves that a certain notion in the relevant ``jeu de taquin'' is well defined; then the strategy already outlined in Thomas-Yong proves the main result, the Littlewood-Richardson rule for the \(K\)-theory of odd orthogonal Grassmannians. Schubert calculus; Littlewood-Richardson rule; orthogonal Grassmannian; jeu de taquin Clifford, Edward; Thomas, Hugh; Yong, Alexander, K-theoretic Schubert calculus for OG\((n, 2n+ 1)\) and jeu de taquin for shifted increasing tableaux, J. Reine Angew. Math., 690, 51-63, (2014) Classical problems, Schubert calculus, Combinatorial aspects of representation theory \(K\)-theoretic Schubert calculus for \(OG(n,2n+1)\) and jeu de taquin for shifted increasing tableaux

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We associate a polynomial to any diagram of unit cells in the first quadrant of the plane using Kohnert's algorithm for moving cells down. In this way, for every weak composition one can choose a cell diagram with corresponding row-counts, with each choice giving rise to a combinatorially-defined basis of polynomials. These \textit{Kohnert bases} provide a simultaneous generalization of Schubert polynomials and Demazure characters for the general linear group. Using the monomial and fundamental slide bases defined earlier by the authors, we show that Kohnert polynomials stabilize to quasisymmetric functions that are nonnegative on the fundamental basis for quasisymmetric functions. For initial applications, we define and study two new Kohnert bases. The elements of one basis are conjecturally Schubert-positive and stabilize to the skew-Schur functions; the elements of the other basis stabilize to a new basis of quasisymmetric functions that contains the Schur functions. Schubert polynomials; Demazure characters; key polynomials; fundamental slide polynomials Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Symmetric functions and generalizations Kohnert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K[X_d,Y_d]=K[x_1,\ldots,x_d,y_1,\ldots,y_d]\) be the polynomial algebra in \(2d\) variables over a field \(K\) of characteristic 0 and let \(\delta\) be the derivation of \(K[X_d,Y_d]\) defined by \(\delta(y_i)=x_i, \delta(x_i)=0\), \(i=1,\ldots,d\). In 1994, \textit{A.~Nowicki} [Polynomial derivations and their rings of constants. Wydawnictwo Uniwersytetu Mikolaja Kopernika, Torun (1994; Zbl 1236.13023)] conjectured that the algebra \(K[X_d,Y_d]^{\delta}\) of constants of \(\delta\) is generated by \(X_d\) and \(x_iy_j-y_ix_j\) for all \(1\leq i<j\leq d\). The affirmative answer was given by several authors using different ideas. In the present paper we give another proof of the conjecture based on representation theory of the general linear group \(\mathrm{GL}_2(K)\). Derivations and commutative rings, Actions of groups on commutative rings; invariant theory, \(T\)-ideals, identities, varieties of associative rings and algebras, Derivations, actions of Lie algebras, Representations of finite symmetric groups, Group actions on affine varieties Another proof of the Nowicki conjecture

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0731.00008.] In this paper the Schur \(S\)-polynomials and the Schur \(Q\)-polynomials are studied. These polynomials are then applied to elimination theory, and Schubert calculus for Grassmannians of isotropic subspaces. Schur polynomials; elimination theory; Schubert calculus for Grassmannians Pragacz, Piotr, Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials.Topics in invariant theory, Paris, 1989/1990, Lecture Notes in Math. 1478, 130-191, (1991), Springer, Berlin Grassmannians, Schubert varieties, flag manifolds Algebro-geometric applications of Schur \(S\)- and \(Q\)-polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Based on Thomas and Yong's \(K\)-theoretic jeu de taquin algorithm, we prove a uniform Littlewood-Richardson rule for the \(K\)-theoretic Schubert structure constants of all minuscule homogeneous spaces. Our formula is new in all types. For the main examples of Grassmannians of type A and maximal orthogonal Grassmannians it has the advantage that the tableaux to be counted can be recognized without reference to the jeu de taquin algorithm. \(K\)-theory in geometry, Grassmannians, Schubert varieties, flag manifolds \(K\)-theory of minuscule varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, using Lusztig's Frobenius maps for quantum groups at roots of unity, the authors give a representation-theoretic proof of the Frobenius-split property of Schubert varieties in the generalized flag varieties. This paper makes an important contribution to the geometry and representation theory of flag varieties. quantum groups; Frobenius Burns, D.; Levenberg, N.; Ma'u, S.; Revesz, Sz., Monge-Ampère measures for convex bodies and Bernstein-Markov type inequalities, Trans. Amer. Math. Soc., 362, 6325-6340, (2010) Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups, Quantum groups (quantized enveloping algebras) and related deformations, Cohomology theory for linear algebraic groups Algebraization of Frobenius splitting via quantum groups.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give closed combinatorial product formulas for Kazhdan-Lusztig polynomials and their parabolic analogue of type \(q\) in the case of Boolean elements, introduced in [\textit{M. Marietti}, J. Algebra 295, No. 1, 1--26 (2006; Zbl 1097.20035)], in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of Boolean elements. Coxeter groups; Kazhdan-Lusztig polynomials; Boolean elements; Poincaré polynomials Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Reflection and Coxeter groups (group-theoretic aspects), Classical problems, Schubert calculus, Hecke algebras and their representations Kazhdan-Lusztig polynomials of Boolean elements

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The manifold \(\mathcal Fl_n\) of complete flags in the \(n\) dimensional vector space \(\mathbb C^n\) over the complex numbers is an object that, by its various definitions, is an object in the intersection of algebra and geometry. On the one hand it can be expressed as the quotient \(B\backslash \text{GL}_n\) of all invertible \(n\times n\)-matrices by its subgroup of lower triangular matrices, and on the other hand as fibers of certain bundles constructed universally from complex vector bundles. Combinatorics enter into the study via the cohomology ring \(H^\ast(\mathcal Fl_n) =H^\ast(\mathcal Fl_n;\mathbb Z)\) with integer coefficients, that can be described as the quotient of the polynomial ring \(\mathbb Z[x_1,\dots, x_n]\) modulo the ideal generated by all non-constant homogeneous functions invariant under permutations of \(x_1,\dots, x_n\). This ring is a free abelian group of rank \(n!\) with basis given by monomials dividing \(\prod_{i=1}^{n-1}x_i^{n-i}\). The ring also has a much more geometric basis given by the Schubert classes \([X_w]\) in the cohomology ring \(H^\ast(\mathcal Fl_n)\). This article makes an important contribution to bridging the algebra and combinatorics of Schubert polynomials with the geometry of Schubert varieties. It brings new perspectives to problems in commutative algebra concerning ideals generated by minors of generic matrices, and provides a geometric context in which polynomial representatives for Schubert classes are uniquely singled out with no choices but a Borel subgroup of the general linear group \(\text{GL}_n\mathbb C\) in such a way that it is geometrically obvious that these representatives have nonnegative coefficients. One of the main ideas in the article is to translate ordinary cohomological statements concerning Borel orbit closures on the flag manifold \(\mathcal Fl_n\) into equivariant-cohomological statements concerning double Borel orbit closures on the \(n\times n\) matrices \(M_n\). To be more precise, the preimage \(\widetilde X_w\subseteq \text{GL}_n\mathbb C\) of a Schubert variety \(X_w\in \mathcal Fl_n\) is an orbit closure for the action of the product \(B\times B^+\) of the lower and upper triangular subgroups of \(\text{GL}_n\mathbb C\) acting by multiplication on the left and right. When \(\overline X_w\subseteq M_n\) is the closure of \(\widetilde X_w\) and \(T\) is the torus in \(B\), the \(T\)-equivariant cohomology class \([\overline X_w]_T\in H_T^\ast(M_n)\) is the polynomial representative. It has positive coefficients because there is a \(T\)-equivariant flat (Gröbner) degeneration of \(\overline X_w\) to \(\mathcal L_w\) that is a union of coordinate subspaces \(L\subseteq M_n\). Each subspace \(L\subseteq \mathcal L_w\) has an equivariant cohomology class \([L]_T\in H_T^\ast(M_n)\) that is a monomial in \(x_1,\dots, x_n\), and the sum of these is \([\overline X_w]_T\). The formula is \([\overline X_w]_T = [\mathcal L_w]_T =\sum_{L\in \mathcal L_w}[L]_T\). More importantly, the authors identify a particularly natural degeneration of the matrix Schubert variety \(\overline X_w\) with a reduced and Cohen-Macaulay limit \(\mathcal L_w\) in which the subspaces have combinatorial interpretations and coincides with known combinatorial formulas for Schubert polynomials. Instead of using equivariant classes associated to closed subvarieties of non-compact spaces the authors develop their theory in the context of multidegrees. The equivariant considerations for matrix Schubert varieties \([\overline X_w]\subseteq M_n\) are then done as multigraded commutative algebra for the Schubert determinantal ideals \(I_w\) cutting out the varieties \(\overline X_w\). The Gröbner geometry of Schubert polynomials introduced provides a geometric explanation for the naturality of Schubert polynomials and their associated combinatorics. In fact the authors describe, for every matrix Schubert variety \(\overline X_w\); (1) its multidegree and Hilbert series in terms of Schubert and Grothendieck polynomials (2) a Gröbner basis consisting of minors in its defining ideal \(I_w\) (3) the Stanley-Reisner complex \(\mathcal L_w\) of its initial ideal \(J_w\), which they prove is Cohen-Macaulay (4) an inductive irredundant algorithm of weak Bruhat order for listing the facets of \(\mathcal L_w\). The authors introduce a powerful inductive method that they call Bruhat induction, for working with determinantal ideals and their initial ideals. Bruhat induction as well as the derivation of the main theorems concerning Gröbner geometry rely on results concerning positivity of torus-equivariant cohomology classes represented by subschemes and shellability of certain simplicial complexes that reflect the nature of reduced subwords of words in Coxeter generators for Coxeter groups. The latter technique gives a new perspective, from simplicial topology, of the combinatorics of Schubert and Grothendieck polynomials. Among the most important applications of the work is the geometrically positive formulae for Schubert polynomials, and connections with Fulton's theory of degeneracy loci, relations between multidegrees and \(K\)-polynomials on \(n\times n\) matrices with classical cohomological theories on the flag manifold, and comparisons with the commutative algebra of determinantal ideals. Stanley-Reisner complex; Coxeter group; Bruhat order; Cohen-Macaulay ideal; initial ideal; Bruhat group; equivariant cohomology; divided differences; Bruhat induction Knutson, [Knutson and Miller 05] A.; Miller, E., Gröbner Geometry of Schubert Polynomials., Ann. Math. (2), 161, 3, 1245-1318, (2005) Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Linkage, complete intersections and determinantal ideals, Classical problems, Schubert calculus, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Group actions on posets and homology groups of posets [See also 06A09] Gröbner geometry of Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use incidence relations running in two directions in order to construct a Kempf-Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott-Samelson resolutions. The second process led to the introduction of \(W\)-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian. flag; Grassmannian; resolution; Schubert; variety Global theory and resolution of singularities (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Bioriented flags and resolutions of Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A previous result of the authors with \textit{P.-E. Chaput} and \textit{N. Perrin} [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 3, 477--494 (2013; Zbl 1282.14016)] states that the closure of the union of all rational curves of fixed degree passing through a Schubert variety in a homogeneous space \(G/P\) is again a Schubert variety. In this paper, we identify this Schubert variety explicitly in terms of the Hecke product of Weyl group elements. We apply our result to give an explicit formula for any two-point Gromov-Witten invariant as well as a new proof of the quantum Chevalley formula and its equivariant generalization. We also recover a formula for the minimal degree of a rational curve between two given points in a cominuscule variety. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Curve neighborhoods of Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The ring of symmetric functions carries the structure of a Hopf algebra. When computing the coproduct of complete symmetric functions \(h_{\lambda}\) one arrives at weighted sums over reverse plane partitions (RPP) involving binomial coefficients. Employing the action of the extended affine symmetric group at fixed level \(n\) we generalise these weighted sums to cylindric RPP and define cylindric complete symmetric functions. The latter are shown to be \(h\)-positive, that is, their expansions coefficients in the basis of complete symmetric functions are non-negative integers. We state an explicit formula in terms of tensor multiplicities for irreducible representations of the generalised symmetric group. Moreover, we relate the complete symmetric functions to a 2D topological quantum field theory (TQFT) that is a generalisation of the celebrated \(\widehat{\mathfrak{sl}}_n\)-Verlinde algebra or Wess-Zumino-Witten fusion ring, which plays a prominent role in the context of vertex operator algebras and algebraic geometry. reverse plane partitions; symmetric functions; topological quantum field theory; Verlinde algebra; generalised symmetric group Symmetric functions and generalizations, Topological quantum field theories (aspects of differential topology), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Hopf algebras and their applications Cylindric reverse plane partitions and 2D TQFT

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X=G/P\) be a homogeneous space and \(\epsilon_k\) the homology class of a simple coroot. For almost all \(X\), the variety \(Z_k(X)\) of degree \(\epsilon_k\) pointed lines in \(X\) is known to be homogeneous. For these \(X\), we show that the 3-point, genus 0, equivariant \(K\)-theoretic Gromov-Witten invariants of lines of degree \(\epsilon_k\) equal quantities obtained in the (ordinary) equivariant \(K\)-theory of \(Z_k(X)\). We apply this to compute the Schubert structure constants \(N_{u,v}^{w,\epsilon_k}\) in the equivariant quantum \(K\)-theory ring of \(X\). Using geometry of spaces of lines through Schubert or Richardson varieties we prove vanishing and positivity properties of \(N_{u,v}^{w,\epsilon_k}\). This generalizes many results from \(K\)-theory and quantum cohomology of \(X\) and gives new identities among the structure constants in the equivariant \(K\)-theory of \(X\). Li, C.; Mihalcea, L. C., \textit{K}-theoretic Gromov-Witten invariants of lines in homogeneous spaces, Int. Math. Res. Not., 2014, 17, 4625-4664, (2014) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry \(K\)-theoretic Gromov-Witten invariants of lines in homogeneous spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of \textit{P. Orlik} and \textit{L. Solomon} [Invent. Math. 56, 167--189 (1980; Zbl 0432.14016)] relating the characteristic polynomial with the ranks of the cohomology of the complement of the arrangement. We also show that the characteristic polynomial can be computed from the total Chern class of the complement of the arrangement. In the case of free arrangements, we prove that this Chern class agrees with the Chern class of the dual of a bundle of differential forms with logarithmic poles along the hyperplanes in the arrangement; this follows from the work of \textit{M. Mustaţǎ} and \textit{H. K. Schenck} [J. Algebra 241, No. 2, 699--719 (2001; Zbl 1047.14007)]. We conjecture that this relation holds for any locally quasi-homogeneous free divisor. We give an explicit relation between the characteristic polynomial of an arrangement and the Segre class of its singularity (``Jacobian'') subscheme. This gives a variant of a recent result of \textit{M. Wakefield} and \textit{M. Yoshinaga} [Math. Res. Lett. 15, No. 4, 795--799 (2008; Zbl 1158.14044)], and shows that the Segre class of the singularity subscheme of an arrangement together with the degree of the arrangement determines the ranks of the cohomology of its complement. We also discuss the positivity of the Chern classes of hyperplane arrangements: we give a combinatorial interpretation of this phenomenon, and discuss the cases of generic and free arrangements. Aluffi, P., Grothendieck classes and Chern classes of hyperplane arrangements, Int. Math. Res. Not. IMRN, 2013, 1873-1900, (2013) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Configurations and arrangements of linear subspaces, Characteristic classes and numbers in differential topology, Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Grothendieck classes and Chern classes of hyperplane arrangements

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The following is a classic open problem in the Schubert calculus of the flag variety: Given any reasonably nice subvariety \(Y \subset \mathrm{Flags}(\mathbb C^n)\), express the homology class of \(Y\) as an integral linear combination of Schubert classes. In this paper, the authors consider the case where \(Y\) is the Peterson variety. By analyzing the cellular structure of the Peterson variety and the group action of a one-dimensional torus on this variety, they reduce the computations in the intersection theory of the flag variety to a systematic combinatorial analysis of the elements of the symmetric group. In the process, they give a partial solution to the first problem introduced above. their proof counts the points of intersection between certain Schubert varieties in the full flag variety and the Peterson variety, and shows that these intersections are proper and transverse. Schubert calculus; intersection theory; Peterson variety Insko, Erik, Schubert calculus and the homology of the Peterson variety, Electron. J. Combin., 22, 2, (2015) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Classical real and complex (co)homology in algebraic geometry Schubert calculus and the homology of the Peterson variety

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the Bernstein polynomial of a family \((W_ y)_{y \in Y}\) of hypersurfaces of \(\mathbb{C}^ n\) with isolated singularities when the family is parametrized by a smooth space \(Y\) and defined by an holomorphic mapping \(F\) on \(\mathbb{C}^ n \times Y\). We show that the partition defined by the Bernstein's polynomial \(b_ y\) of the fiber \(W_ y\) of the parameter space \(Y\) is locally finite and ``constructible''. We prove the existence of a generic Bernstein polynomial which coincide with the Bernstein polynomial of the generic fiber. In the particular case where the family \((W_ y)_{y \in Y}\) has a constant Milnor number we establish the existence of a ``good operator in \(s\)'' in the ring of the relative differential operators which are polynomials in \(s\), nullifying \(F^ s\); and the existence of a ``relative'' Bernstein polynomial. Bernstein polynomial; hypersurfaces; isolated singularities; Milnor number Joël Briançon, Françoise Geandier & Philippe Maisonobe, ``Déformation d'une singularité isolée d'hypersurface et polynômes de Bernstein'', Bull. Soc. Math. Fr.120 (1992) no. 1, p. 15-49 Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties Deformation of an isolated singularity of a hypersurface and Bernstein polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the Hilbert series of the complex Grassmannian using invariant theoretic methods. This is made possible by showing that the denominator of the \(q\)-Hilbert series is a Vandermonde-like determinant. We show that the \(h\)-polynomial of the Grassmannian coincides with the \(k\)-Narayana polynomial. A simplified formula for the \(h\)-polynomial of Schubert varieties is given. Finally, we use a generalized hypergeometric Euler transform to find simplified formulae for the \(k\)-Narayana numbers, i.e. the \(h\)-polynomial of the Grassmannian. Hilbert series of the Grassmannian; Narayana numbers; Euler's hypergeometric transform Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, \(q\)-calculus and related topics, Actions of groups on commutative rings; invariant theory, Grassmannians, Schubert varieties, flag manifolds, Applications of hypergeometric functions Hilbert series of the Grassmannian and \(k\)-Narayana numbers

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Starting from a Kac-Moody algebra and an associated group G with Tits- system(B,H) we investigate the geometric and topological structure of the generalized flag manifold G/B, extending results of \textit{A. Borel} [Ann. Math., II. Ser. 57, 115-207 (1953; Zbl 0052.400)], \textit{I. N. Bernstein, I. M. Gel'fand} and \textit{S. I. Gel'fand} [Russ. Math. Surveys 28, 1-26 (1973); translation from Usp. Mat. Nauk 28, No.3(171), 3-26 (1973; Zbl 0286.57025)] and \textit{M. Demazure} [Ann. Sci. Éc. Norm. Super., IV. Ser. 7, 53-88 (1974; Zbl 0312.14009)] in the finite-dimensional case, e.g. generation of the homology by Schubert cycles, characteristic homomorphism, evaluation of polynomials on the Cartan algebra as cohomology classes on Schubert cycles. Detailed proofs of these results are going to appear elsewhere. Further generalizations have meanwhile been obtained by Kac, Peterson, and Kumar (unpublished). Kac-Moody algebras; flag manifolds; Schubert cycles; Weyl group invariants Eugene Gutkin and Peter Slodowy, Cohomologie des variétés de drapeaux infinies, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 15, 625 -- 627 (French, with English summary). Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties, Group actions on varieties or schemes (quotients) Cohomologie des variétés de drapeaux infinies

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use the recently introduced \textit{padded Schubert} polynomials to prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is \(\binom{n}{2}!\) for both the \textit{code weights} and the \textit{Chevalley weights}, generalizing a result of Stembridge. We also define weights which give a one-parameter family of strong order analogues of Macdonald's well-known reduced word identity for Schubert polynomials. Algebraic aspects of posets, Classical problems, Schubert calculus Weighted enumeration of Bruhat chains in the symmetric group

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov-Witten invariants in the multiplication table of the Schubert classes are nonnegative and deduce Golyshev's conjecture \(\mathcal{O}\) holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin's conjecture, an exceptional collection of maximal length in the derived category. quasi homogeneous varieties; quantum cohomology; Dubrovin's conjecture Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Fano varieties The small quantum cohomology of the Cayley Grassmannian

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using a blend of combinatorics and geometry, we give an algorithm for algebraically finding all flags in any zero-dimensional intersection of Schubert varieties with respect to three transverse flags, and more generally, any number of flags. In particular, the number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold. Our algorithm is based on solving a limited number of determinantal equations for each intersection (far fewer than the naive approach). These equations may be used to compute Galois and monodromy groups of intersections of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of \textit{K. Eriksson} and \textit{S. Linusson} [Adv. Appl. Math. 25, No. 2, 194--211 (2000; Zbl 0957.05112); ibid. 212--227 (2000; Zbl 0957.05113)], and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a Littlewood-Richardson coefficient to be 0. We describe pathologies of Eriksson and Linusson's permutation array varieties (failure of existence, irreducibility, equidimensionality, and reducedness of equations), and define the more natural permutation array schemes. In particular, we give several counterexamples to the Realizability Conjecture based on classical projective geometry. Finally, we give examples where Galois/monodromy groups experimentally appear to be smaller than expected. Sara Billey and Ravi Vakil, Intersections of Schubert varieties and other permutation array schemes, Algorithms in algebraic geometry, IMA Vol. Math. Appl., vol. 146, Springer, New York, 2008, pp. 21 -- 54. Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Intersections of Schubert varieties and other permutation array schemes

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the quantum Witten-Kontsevich series introduced by \textit{A. Buryak} et al. [Int. Math. Res. Not. 2020, No. 24, 10381--10446 (2020; Zbl 1464.37071)] as the logarithm of a quantum tau function for the quantum KdV hierarchy. This series depends on a genus parameter \(\epsilon\) and a quantum parameter \(\hbar\). When \(\hbar=0\), this series restricts to the Witten-Kontsevich generating series for intersection numbers of psi classes on moduli spaces of stable curves. We establish a link between the \(\epsilon=0\) part of the quantum Witten-Kontsevich series and one-part double Hurwitz numbers. These numbers count the number of nonequivalent holomorphic maps from a Riemann surface of genus \(g\) to \(\mathbb{P}^1\) with a complete ramification over \(0\), a prescribed ramification profile over \(\infty\) and a given number of simple ramifications elsewhere. \textit{I. P. Goulden} et al. [Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)] proved that these numbers have the property of being polynomial in the orders of ramification over \(\infty\). We prove that the coefficients of these polynomials are the coefficients of the quantum Witten-Kontsevich series. We also present some partial results about the full quantum Witten-Kontsevich power series. moduli space of curves; double ramification cycle; quantum KdV; quantum tau function; Hurwitz numbers Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Deformation quantization, star products, Families, moduli of curves (algebraic) The quantum Witten-Kontsevich series and one-part double Hurwitz numbers

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We characterize by pattern avoidance the Schubert varieties for \(\mathrm{GL}_n\) which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighborhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. One application is a new formula for certain specializations of Schubert polynomials. Schubert varieties; permutation patterns Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds Which Schubert varieties are local complete intersections?

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is organized as follows. In Section 1 we describe our results. Section 2 contains necessary background. Section 3 contains most of our combinatorial analysis. In Section 4, we study the effect on cohomology of certain maps between flag manifolds and compute specializations of the variables in a Schubert polynomial. In Section 5, we prove the identities when \({\mathfrak S}_v\) is a Schur polynomial. In Section 6, we use these identities to compute many of the \(c^w_{uv}\). Young tableaux; combinatorial interpretation; Bruhat order; Littlewood-Richardson coefficients; Grassmannian permutations; cohomology; flag manifolds; Schubert polynomial; Schur polynomial N. Bergeron and F. Sottile. ''Schubert polynomials, the Bruhat order, and the geometry of flag manifolds''. Duke Math. J. 95 (1998), pp. 373--423.DOI. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Schubert polynomials, the Bruhat order, and the geometry of flag manifolds

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following the work of Okounkov and Pandharipande, Diaconescu and the recent work of Ciocan-Fontanine et al. studying the equivariant quantum cohomology \(QH^{*}_{(\mathbb{C}^{*})^{2}}(\text{Hilb}_{n})\) of the Hilbert scheme and relative Donaldson-Thomas theory of \(\mathbb{P}^{1}\times \mathbb{C}^{2}\), the authors establish a connection between \(J\)-function of the Hilbert scheme and a certain combinatorial identity in two variables. The authors also generalize this identity to a multivariable identity. Hilbert scheme of points; quantum cohomology; hook walk; hook-length formula Ciocan-Fontanine, I.; Konvalinka, M.; Pak, I., Quantum cohomology of hilb\_{}\{n\}(\( \mathbb{C} \)\^{}\{2\}) and the weighted Hood walk on the Young diagrams, J. Alg., 349, 268, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Parametrization (Chow and Hilbert schemes) Quantum cohomology of Hilb\(_n(\mathbb C^2)\) and the weighted hook walk on Young diagrams

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Pieri's formula asserts that the product of a Schubert class and a special Schubert class is a sum of certain other Schubert classes, each with coefficient 1. This determines the multiplicative structure of the Chow ring of a Grassmann variety. Pieri's formula also arises in algebra, combinatorics, and representation theory, and has several proofs in these contexts. We present a new geometric proof of Pieri's formula, explicitly describing a sequence of deformations (inducing rational equivalence) that transform a general intersection of a Schubert variety with a special Schubert variety into a union of distinct Schubert varieties. This gives an understanding of the structure of rational equivalence on Grassmann varieties in terms of the combinatorics of the Bruhat order of the Schubert cellular decomposition. This proof enables one to determine some enumerative problems [\textit{F. Sottile}, J. Pure Appl. Algebra 117-118, 601-615 (1997; Zbl 0889.14026); section 5] without reference to a Chow or cohomology ring, the traditional tool in enumerative geometry. Moreover, these deformations show that these enumerative problems may be solved over the real numbers [the author, loc. cit.]. The geometry of these deformations is quite interesting and their form parallels a combinatorial proof of Pieri's formula [see: \textit{W. Fulton}, ``Young tableaux. With applications to representation theory and geometry'', Lond. Math. Soc. Stud. Texts 35 (1997; Zbl 0878.14034); p. 24] using Schensted insertion [\textit{C. Schensted}, Can. J. Math. 13, 179-191 (1961; Zbl 0097.25202)]. -- Their explicit nature leads to homotopy continuation algorithms for finding numerical solutions to enumerative problems involving any number of special Schubert conditions. deformations of Schubert variety; Pieri's formula; rational equivalence; Grassmann varieties; enumerative problems Sottile, F.: Pieri's Formula via explicit rational equivalence. Canad. J. Math. 49, 1281--1298 (1997) Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Enumerative problems (combinatorial problems) in algebraic geometry Pieri's formula via explicit rational equivalence

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Kronecker coefficient \(g_{\lambda \mu \nu}\) is the multiplicity of the \(\mathrm{GL}(V) \times \mathrm{GL}(W)\)-irreducible \(V_\lambda \otimes W_\mu\) in the restriction of the \(\mathrm{GL}(X)\)-irreducible \(X_\nu\) via the natural map \(\mathrm{GL}(V ) \times \mathrm{GL}(W ) \to \mathrm{GL}(V \otimes W)\), where \(V\), \(W\) are \(\mathbb C\)-vector spaces and \(X =V \otimes W\) . A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality. Using these nonstandard objects as a guide, we follow the approach of \textit{B. Adsul}, \textit{M. Sohoni} and \textit{K. V. Subrahmanyam} [``Quantum deformations of the restriction of \(\mathrm{GL}_{mn}(\mathbb{C})\)-modules to \(\mathrm{GL}_m(\mathbb{C}) \times \mathrm{GL}_n(\mathbb{C})\)'', Preprint, \url{arXiv:0905.0094}] to construct, in the case \(\dim(V ) = \dim(W ) = 2\), a representation \(\check X_\nu\) of the nonstandard quantum group that specializes to \(\mathrm{Res}_{\mathrm{GL}(V ) \times \mathrm{GL}(W)} X_\nu\) at \(q = 1\). We then define a global crystal basis \(+\mathrm{HNSTC}(\nu)\) of \(\check X_\nu\) that solves the two-row Kronecker problem: the number of highest weight elements of \(+\mathrm{HNSTC}(\nu)\) of weight \((\lambda, \mu)\) is the Kronecker coefficient \(g_{\lambda \mu \nu}\). We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the \(U_q (\mathfrak{sl}_2)\) graphical calculus and use this to organize the crystal components of \(+\mathrm{HNSTC}(\nu)\) into eight families. This yields a fairly simple, positive formula for two-row Kronecker coefficients, generalizing a formula of \textit{A. A. H. Brown} et al. [Pac. J. Math. 248, No. 1, 31--48 (2010; Zbl 1230.05275)]. As a byproduct of the approach, we also obtain a rule for the decomposition of \(\mathrm{Res}_{\mathrm{GL}_2 \times \mathrm{GL}_2 \times S_2} X_\nu\) into irreducibles. Kronecker problem; complexity theory; canonical basis; quantum group; Hecke J. Blasiak, K. Mulmuley & M. Sohoni (2015). Geometric complexity theory IV: nonstandard quantum group for the Kronecker problem. \textit{Memoirs of the AMS}\textbf{235}(1109). Hecke algebras and their representations, Quantum groups (quantized enveloping algebras) and related deformations, Quantum groups (quantized function algebras) and their representations, Representation theory for linear algebraic groups, Representations of finite symmetric groups, Combinatorial aspects of representation theory, Geometric invariant theory, Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.), Complexity of computation (including implicit computational complexity) Geometric complexity theory. IV: Nonstandard quantum group for the Kronecker problem.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors give a simple necessary and sufficient condition for a Schubert variety \(X_w\), to be smooth when \(w\) is a freely braided element of a simply laced Weyl group; such elements were introduced by the authors in a previous work [Ann. Comb. 6, No. 3--4, 337--348 (2002; Zbl 1052.20028)]. This generalizes in one direction a result of \textit{C.~K.~Fan} [Transform. Groups 3, No. 1, 51--56 (1998; Zbl 0912.20033)] concerning varieties indexed by short-braid avoiding elements. They also derive generating functions for the freely braided elements that index smooth Schubert varieties. All results are stated and proved only for the simply laced case. Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects) Schubert varieties and free braidedness

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we give combinatorial formulas for the Hilbert coefficients, \(h\)-polynomial and the Cohen-Macaulay type of Schubert varieties in Grassmannians in terms of the posets associated with them. As a consequence, necessary conditions for a Schubert variety to be a complete intersection and combinatorial criteria are given for a Schubert variety to be Gorenstein and almost Gorenstein, respectively. Grassmannian; Schubert variety; Hilbert coefficients; Gorenstein ring; almost Gorenstein ring Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Dimension theory, depth, related commutative rings (catenary, etc.), Grassmannians, Schubert varieties, flag manifolds Hilbert coefficients of Schubert varieties in Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau-Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the \(K\)-theory algebra of the partial flag variety. To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the \textit{qKZ} equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and \textit{qKZ} difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit. Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety. Gromov-Witten invariants; quantum differential equations; qKZ difference equations; Yangian action; multidimensional hypergeometric functions Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Structure of families (Picard-Lefschetz, monodromy, etc.), Other hypergeometric functions and integrals in several variables Landau-Ginzburg mirror, quantum differential equations and \textit{qKZ} difference equations for a partial flag variety

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper is devoted to the purely category theoretic background of Grothendieck's six functor formalism with a view towards applications in algebraic geometry and algebraic topology. The basic theory is developed without assuming the categories to be triangulated, so the results hold both before and after passing to derived categories. The general setup considered in the paper is two pairs of adjoint functors between closed symmetric monoidal categories, which are suggestively denoted by \((f^*,f_*)\) and \((f_!,f^!)\), respectively. The main aim of the authors is to clarify which relations among these adjoint pairs and the \(\otimes\)- and Hom-functors can be deduced on a purely formal level, and in which points additional input from the concrete situation is needed. Apart from the general situation (referred to as the Verdier-Grothendieck-context) the authors also consider two special cases, namely \(f_*=f_!\) (the Grothendieck-context), and \(f^*=f^!\) (the Wirthmüller-context). Relations between the functors are then studied using dualizing objects. In the last two sections, the authors pass to the setting of triangulated categories and prove, under additional assumptions on the categories in question, formal versions of the Grothendieck-, the Verdier-, and the Wirthmüller-isomorphism theorems. closed symmetric monoidal category; pair of adjoint functors; triangulated category; Grothendieck duality; Verdier duality; Wirthmüller isomorphism H. Fausk, P. Hu and J.\ P. May, Isomorphisms between left and right adjoints, Theory Appl. Categ. 11 (2003), no. 4, 107-131. Closed categories (closed monoidal and Cartesian closed categories, etc.), Varieties and morphisms, Monoidal categories (= multiplicative categories) [See also 19D23], Derived categories, triangulated categories, Applied homological algebra and category theory in algebraic topology, Categories in geometry and topology Isomorphisms between left and right adjoints

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{S. Barannikov} and \textit{M. Kontsevich} [Int. Math. Res. Not. 1998, No. 4, 201--215 (1998)] constructed a DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra \(\mathbf{t}\) for a compact smooth Calabi-Yau complex manifold \(M\) of dimension \(m\), which gives rise to the \(B\)-side formal Frobenius manifold structure in the homological mirror symmetry conjecture. The cohomology of the DGBV algebra \(\mathbf{t}\) is isomorphic to the total singular cohomology \(H^{\bullet}(M)=\bigoplus_{k=0}^{2m}H^{k}(M,\mathbb{C})\) of \(M\). If \(M=X_{G}(\mathbb{C})\), where \(X_{G}\) is the hypersurface defined by a homogeneous polynomial \(G(\underline{x})\) in the projective space \(\mathbb{P}^{n}\), then we give a purely algorithmic construction of a DGBV algebra \({\mathcal{A}}_{U}\), which computes the primitive part \(\bigoplus_{k=0}^{m}\mathbf{PH}^{k}\) of the middle-dimensional cohomology \(\bigoplus_{k=0}^{m}H^{k}(M,\mathbb{C})\), using the de Rham cohomology of the hypersurface complement \(U_{G}:=\mathbb{P}^{n}\setminus X_{G}\) and the residue isomorphism from \(H_{\text{dR}}^{k}(U_{G}/\mathbb{C})\) to \(\mathbf{PH}^{k}\). We observe that the DGBV algebra \({\mathcal{A}}_{U}\) still makes sense even for a singular projective Calabi-Yau hypersurface, i.e. \({\mathcal{A}}_{U}\) computes \(\bigoplus_{k=0}^{m}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})\) even for a singular \(X_{G}\). Moreover, we give a precise relationship between \({\mathcal{A}}_{U}\) and \(\mathbf{t}\) when \(X_{G}\) is smooth in \(\mathbf{P}^{n}\). Frobenius manifold; homological mirror symmetry conjecture; singular cohomology; de Rham cohomology; Calabi-Yau hypersurface Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Nonabelian homotopical algebra, Mirror symmetry (algebro-geometric aspects), Algebraic geometry methods for problems in mechanics Differential Gerstenhaber-Batalin-Vilkovisky algebras for Calabi-Yau hypersurface complements

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this short paper is to introduce a multi-variable analogue of the Kostka-Shoji polynomial introduced by \textit{T. Shoji} [Sci. China, Math. 61, No. 2, 353--384 (2018; Zbl 1494.05115)]. The generalisation replaces integer partitions \(\lambda,\mu\) with multipartitions with \(r\) components, for \(r>1\). The authors then show that this new version has applications related to Lusztig's iterated convolution diagram for the cyclic quiver \(\tilde A_{r-1}\). This paper is not for the faint-hearted. It is technical right from the start, and the reader is referred to the references for most of the background. There are no examples to aid the reader's understanding. The advantage of all this is that the paper is short, so gives a very concise account for experts. Kostka-Shoji polynomials; cyclic quiver; convolution diagram; Frobenius splitting; affine flag variety; Bott-Samelson-Demazure-Hansen resolution Finkelberg, M; Ionov, A, Kostka-shoji polynomials and lusztig's convolution diagram, (2016) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Kostka-Shoji polynomials and Lusztig's convolution diagram

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We discuss a surprising relationship between the partially ordered set of Newton points associated with an affine Schubert cell and the quantum cohomology of the complex flag variety. The main theorem provides a combinatorial formula for the unique maximum element in this poset in terms of paths in the quantum Bruhat graph, whose vertices are indexed by elements in the finite Weyl group. Key to establishing this connection is the fact that paths in the quantum Bruhat graph encode saturated chains in the strong Bruhat order on the affine Weyl group. This correspondence is also fundamental in the work of Lam and Shimozono establishing Peterson's isomorphism between the quantum cohomology of the finite flag variety and the homology of the affine Grassmannian. One important geometric application of the present work is an inequality which provides a necessary condition for nonemptiness of certain affine Deligne-Lusztig varieties in the affine flag variety. Grassmannians, Schubert varieties, flag manifolds, \(p\)-adic cohomology, crystalline cohomology, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Linear algebraic groups over local fields and their integers, Symmetric functions and generalizations Maximal Newton points and the quantum Bruhat graph

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author studies two \(dg\)-operads, dual in the sense of Ginzburg-Kapranov, which are related to the moduli spaces of the title. The algebras described by these operads are, respectively, the author's ``gravity'' algebras [Commun. Math. Phys. 163, No. 3, 473-489 (1994; Zbl 0806.53073)] and the algebras discovered by \textit{R. Dijkgraaf}, \textit{H. Verlinde} and \textit{E. Verlinde} [Nuclear Phys. B 352, No. 1, 59-86 (1991)], which the author rechristens ``polycommutative''. The latter have a sequence of operations \(A^{\otimes n} \to A\) satisfying an appropriate generalization of associativity. An important class of examples is provided by the quantum cohomology of compact Kähler manifolds. As in much other recent work, a key role is played by the moduli spaces \({\mathcal M}_{0,n}\) of \(n\)-punctured Riemann spheres and the Knudsen-Deligne-Mumford compactification \(\overline {\mathcal M}_{0,n}\). The relevant technical tools include the spectral sequence of the natural stratification of the compactified moduli space and mixed Hodge theory which for genus 0 is pure. The author is building on much of his earlier work, especially that with Kapranov, cf. cyclic and modular operads. Graphs and trees play a significant part, although the given combinatorial definition of graph is far from perspicuous. He also obtains new formulas for the characters of the homology of \({\mathcal M}_{0, n}\) and of \(\overline {\mathcal M}_{0,n}\) as \({\mathcal S}_n\)-modules. \(dg\)-operads; gravity algebras; polycommutative; punctured Riemann spheres; stratification of compactified moduli space; homology characters; moduli spaces; quantum cohomology; compact Kähler manifolds; Knudsen-Deligne-Mumford compactification; spectral sequence; mixed Hodge theory Getzler, E.: Operads and moduli spaces of genus \(0\) Riemann surfaces. In Dijkgraaf, R., Faber, C., van der Gerr, G. (eds.) The Moduli Space of Curves, volume 129 of \textit{Progress in Mathematics}, pp. 199-230. Birkhäuser, Basel (1995) Monoidal categories (= multiplicative categories) [See also 19D23], Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Homological algebra in category theory, derived categories and functors, Applications of differential geometry to physics, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Other \(n\)-ary compositions \((n \ge 3)\), Quantization in field theory; cohomological methods Operads and moduli spaces of genus 0 Riemann surfaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author provides an alternative proof of the ``component formula'' of Knutson, Miller and Shimozono, replacing the ``Gröbner degeneration part'' of their original argument by a combinatorial result (Theorem 1 in the paper). Moreover, he outlines a geometric project suggested by the striking correspondence between the ``splitting'' formula for the ordinary Schubert polynomials of Lascoux and Schützenberger in terms of quiver coefficients (stated as Theorem 2 in the paper), and an analogous formula (Theorem 3 in the paper) for the BCD-Schubert polynomials of Billey and Haiman in terms of positive combinatorial coefficients. degeneracy loci; quiver polynomials; generalized Littlewood--Richardson coefficients; Schubert polynomials Yong, A.: On combinatorics of quiver component formulas. J. Algebr. Comb. 21, 351--371 (2005) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus On combinatorics of quiver component formulas

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a finite-dimensional vector field over an arbitrary ground field. In the paper under review a description of pairs of subspaces in \(V\) of fixed dimensions \(k\) and \(l\) up to the action of the group \(B\subset\text{GL}(V)\) of nondegenerate upper-triangular matrices is obtained. In other words, the author describes \(B\)-orbits in the direct product \(X=\text{Gr}(k,V)\times \text{Gr}(l,V)\) of two Grassmannians. This description is similar to the Schubert decomposition for Grassmannians or to the Ehresmann-Bruhat decomposition for full flag varieties. The combinatorial description of \(B\)-orbits in \(X\) may be found in [\textit{P.~Magyar, J.~Weyman} and \textit{A.~Zelevinsky}, Adv. Math. 141, No. 1, 97--118 (1999; Zbl 0951.14034)]; the proof there is based on quiver theory. The description given in the present paper uses only elementary linear algebra. The closures of \(B\)-orbits in \(X\) may be viewed as analogues of Schubert varieties in Grassmannians. The singularities of Schubert varieties admit nice desingularizations constructed by Bott and Samelson. In this paper, the author constructs a similar desingularization for the closures of \(B\)-orbits in \(X\). Grassmannian; spherical variety; desingularization Grassmannians, Schubert varieties, flag manifolds, Vector spaces, linear dependence, rank, lineability Desingularizations of Schubert varieties in double Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Grothendieck--Teichmüler group \(\widehat{GT}\) is a certain abstractly constructed profinite group in which the absolute Galois group \(G_{\mathbb Q}\) of the rationals is naturally embedded. It is not known whether the embeding is actually an isomorphism. In this paper, the author derives a formula which the image of \(G_{\mathbb Q}\) satisfies. It is not known if the formula holds for \(\widehat{GT}\). Elements \(\sigma\) of \(\widehat{GT}\) give rise to elements \(f_\sigma\) in the free two generator profinite group. Let the ring \(R\) be the inverse limit of the rings \(\mathbb Z[X,Y]/(X^n-1, Y^n-1)\) as \(n \to \infty\) multiplicatively. Let \(x\) and \(y\) be the limits of \(X\) and \(Y\) in \(R\). The author finds an identity satisfied in \(GL_2(R)\) for \(f_\sigma(A,B)\) where \(A\) (\(B\)) is the upper (lower) triangular matrix with off diagonal \(1-x\) (\(1-y\)) and diagonal \(1,x\) (\(y,1\)). Specializing \(x\) and \(y\) to roots of unity in \(\mathbb Q_{ab}\) then yields various explicit matrix equations that the \(f_\sigma\) must satisfy. The author's formula which the image of \(G_{\mathbb Q}\) satisfies is also phrased in terms of the \(f_\sigma\)'s. absolute Galois group of the rationals Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Galois theory Some classical views on the parameters of the Grothendieck-Teichmüller group

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors use the construction of mirror pairs proposed by \textit{P. Berglund} and \textit{T. Hübsch} [Nucl. Phys. B 393 (1993; Zbl 1245.14039)]. Let \(X_W\) be a hypersurface in a weighted projective space, defined by a quasihomogeneous polynomial \(W\). Let \(X_{W^T}\) be a hypersurface in other weighted projective space, where \(W^T\) is a quasihomogeneous polynomial obtained by transposing the coefficients matrix of \(W\). For \(G \subset \mathrm{Aut}(\{W=0\})\) and \(G^T \subset \mathrm{Aut}(\{W^T=0\})\) satisfying a certain correspondence, the Calabi-Yau \(X_W/G\) and \(X_{W^T}/G^T\) are expected to be a dual pair. This approach to mirror symmetry allows to describe a vast range of cases which are not covered by a more geometric approach due to Batyrev and Borisov. The duality between \(G\) and \(G^T\) was precisely stated by Berglund and Hübsch only in some cases. However, recently \textit{M. Krawitz} [``FJRW rings and Landau-Ginzburg mirror symmetry'', \url{arXiv:0906.0796}] found a general construction for the dual group \(G^T\) and proved a mirror symmetry theorem for all invertible polynomials \(W\) and all admissible groups \(G\) in the Landau-Ginzburg setting. In the article under review the authors prove that \(X_W/G\) and \(X_{W^T}/G^T\) are a mirror pair of Calabi-Yau orbifolds, i.e. there is a 90 degrees rotation of the Hodge diamond of the Chen-Ruan orbifold cohomology: \[ H^{p,q}_{CR}([X_W/\widetilde{G}]; \mathbb{C}) \simeq H^{N-2-p,q}_{CR}([X_{W^T}/\widetilde{G}^T]; \mathbb{C}). \] This theorem is a direct consequence of the Krawitz's Landau-Ginzburg mirror symmetry theorem and the main theorem of this article: the cohomological Landau-Ginzburg/Calabi-Yau correspondence. It is understood as an isomorphism of the Chen-Ruan orbifold cohomology on the Calabi-Yau side and the state space of Fan-Jarvis-Ruan-Witten theory on the Landau-Ginzburg side: \[ H^{p,q}_{CR}([X_W/\widetilde{G}]; \mathbb{C}) \simeq H^{p,q}_{FJRW}(W,G;\mathbb{C}). \] The proof is accomplished by building a common combinatorial model for both theories. mirror symmetry; Landau-Ginzburg/Calabi-Yau correspondence; orbifold Chen-Ruan cohomology Chiodo, A.; Ruan, Y., \textit{LG/CY correspondence: the state space isomorphism}, Adv. Math., 227, 2157-2188, (2011) Calabi-Yau manifolds (algebro-geometric aspects), Mirror symmetry (algebro-geometric aspects), General geometric structures on manifolds (almost complex, almost product structures, etc.) LG/CY correspondence: the state space isomorphism

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper the authors study how much Schubert calculus can be extended to the real case. They consider a 2-parameter family of enumerative problems over the reals. (see \textit{F. Sottile} [Bull. Am. Math. Soc., New Ser. 47, No. 1, 31--71 (2010; Zbl 1197.14052)]). More precisely, they use the method of \textit{R. Vakil} [Ann. Math. (2) 164, No. 2, 489--512 (2006; Zbl 1115.14043)] for the complex case and extend it to the real case. They calculate the solution function of the 2-parameter family and show that in the even case the solution function is constant modulo 4. (see \textit{N. Hein} et al. [J. Reine Angew. Math. 714, 151--174 (2016; Zbl 1403.14088)], and \textit{N. Hein} et al. [São Paulo J. Math. Sci. 7, No. 1, 33--58 (2013; Zbl 1351.14036)]). real enumerative geometry; enumerative real algebraic geometry; modulo four congruence; real Grassmannian; real Schubert variety; four medials Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Real algebraic sets Real solutions of a problem in enumerative geometry

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials To each function \(r\) from the subsets of \(\{1,2,\dots,k\}\) to integers the authors associate the variety \(X\subset ({\mathbb C}^n)^k\) consisting of the \(k\)-tuples of vectors in \(v_1,v_2,\dots,v_k\in{\mathbb C}^n\) having the property: \(\mathrm{rank}(\mathrm{lin}(v_{i_1},v_{i_2},\dots,v_{i_s})=r(\{i_1,i_2,\dots,i_s\})\). The closure of \(X\) is called a matrix matroid variety. The crucial for the paper is the example of the Menelaus configuration. It shows that the ideal defining this particular matrix matroid variety is not generated just by suitable determinants of minors. On the other hand the matrix Schubert varieties, see e.g. [\textit{A. Knutson} and \textit{E. Miller}, Ann. Math. (2) 161, No. 3, 1245--1318 (2005; Zbl 1089.14007)] are easy from that point of view. The main subject of the paper are Thom polynomials (or multidegrees) of matrix matroid varieties. These are the fundamental classes considered in equivariant cohomology with respect to action of the product of the torus \(T=({\mathbb C}^*)^k\) and the linear group \(\mathrm{GL}_n({\mathbb C})\). The formula for the Thom polynomial of the matrix Schubert variety is given. The result follows from the previous work of two authors on double Schubert polynomials [\textit{L. M. Fehér} and \textit{R. Rimányi}, Cent. Eur. J. Math. 1, No. 4, 418--434 (2003; Zbl 1038.57008)]. Except from Menelaus configuration there are also considered other basic examples: Ceva, Pappus and Desargues configurations. Already for these simple configuration the Thom polynomials are complicated, so the authors exhibit the result specializing the torus variables \(d_1,d_2,\dots d_k\in H^*(BT)\) to 0. The interpretations of the Thom polynomial in the spirit of Gromov-Witten is given. Some coefficients of the Thom polynomial have an interpretation in terms of the enumerative geometry. All coefficients are nonnegative when expanded in the Schur basis of \(H^*(\mathrm{GL}_n({\mathbb C}))\) and the monomial basis of \(H^*(BT)\) (with the right signs). The positivity is proven as in [\textit{P. Pragacz} and \textit{A. Weber}, Trends in Mathematics, 117--129 (2008; Zbl 1157.14039)]: the classifying space \(\mathrm{BGL}_n({\mathbb C})\) is approximated by finite Grassmannians and the considered coefficients are equal to intersection numbers of some effective cycles on Grassmannian. The nonnegativity follows by general position argument. The stabilization with respect to the dimension of the ambient space \({\mathbb C}^n\) is discussed. The formulas follows from the general rule based on the Atiyah-Bott localization theorem, as studied in [\textit{L. M. Fehér} and \textit{R. Rimányi}, Ann. Math. (2) 176, No. 3, 1381--1426 (2012; Zbl 1264.32023)]. The computations of the Thom polynomials for the considered examples was done by the combination of interpolation method (loc. cit.) and a version of the restriction method [\textit{L. M. Fehér} and \textit{R. Rimányi}, Contemporary Mathematics 354, 69--93 (2004; Zbl 1074.32008)]. It just so happens that the listed by the authors conditions determine the Thom polynomials here. Thom polynomial; matroid; enumerative geometry; Schubert calculus Fehér, L; Némethi, A; Rimányi, R, Equivariant classes of matrix matroid varieties, Comment. Math. Helv., 87, 861-889, (2012) Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Equivariant classes of matrix matroid varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We outline the results of our revisiting Hermann Schubert's work on the enumerative geometry of cuspidal cubics in \(\mathbb P^3\) [Section 23 of his paper Kalkül der abzählenden Geometrie, Teubner (1879; JFM 11.0460.01), Reprint Springer (1979; Zbl 0417.51008)]. There are three main aspects that we would like to point out. First, we describe the spaces parameterizing cuspidal cubics in \(\mathbb P^3\), as well as several different degenerations, using modern algebraic geometry language and techniques. Then we get formulas, by means of today's intersection theory, for the relevant relations among conditions and degenerations, and for all the intersection numbers in which Schubert was in principle interested. And finally there is the computational aspect, which has been an adventure on its own: the computations have been performed by means of the mathematical computation system OmegaMath, together with the WIT module. They are discussed briefly in the final section, with references to detailed information, and here we would just like to say that one of our motivations has been to test that system in what has turned out to be an interesting computational project. Our final table for the cuspidal cubics, which has the 19778 nonzero numbers involving the nine first-order conditions considered by Schubert, fully confirms the fraction of numbers computed by Schubert, as listed on pages 140--143 of the Kalkül. Miret, J.; Xambó, S.; Hernández, X.: Completing H. Schubert's work on the enumerative geometry of cuspidal cubics in P3. Comm. algebra 31, No. 8, 4037-4068 (2003) Enumerative problems (combinatorial problems) in algebraic geometry, Projective techniques in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Families, moduli of curves (algebraic) Completing Hermann Schubert's work on the enumerative geometry of cuspidal cubics in \(\mathbb P^3\).

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Combining results of \textit{T. K. Lam} [B and D analogues of stable Schubert polynomials and related insertion algorithms. Massachusetts Institute of Technology (PhD Thesis) (1995)] and \textit{J. Stembridge} [Adv. Math. 74, No. 1, 87--134 (1989; Zbl 0677.20012)], the type \(C\) Stanley symmetric function \(F_w^C(\mathbf x)\), indexed by an element \(w\) in the type \(C\) Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements. Stanley symmetric functions; crystal bases; Kráskiewicz insertion; mixed Haiman insertion; unimodal tableaux; primed tableaux Combinatorial aspects of representation theory, Symmetric functions and generalizations, Representations of finite symmetric groups, Classical problems, Schubert calculus Crystal analysis of type \(C\) Stanley symmetric functions

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert calculus on the space of \(d\)-dimensional linear subspaces of a smooth \(n\)-dimensional quadric lying in the complex projective space is the object of study in this article. Following Hodge and Pedoe the author develops the intersection theory of this space in a purely combinatorial manner. It is proved in particular that if a triple intersection of Schubert cells on this space is nonempty then a certain combinatorial relation holds among the Schubert symbols involved, similar to the classical one. The sufficiency of these necessary conditions is also studied. Several examples are examined to illustrate the necessity and sufficiency of these conditions. subspaces of a quadric; Schubert calculus; intersection theory; intersection of Schubert cells Sertöz, S.: A triple intersection theorem for the varieties \(SO(n)/pd\), Fund. math. 142, No. 3, 201-220 (1993) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A triple intersection theorem for the varieties \(SO(n)/P_ d\)

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W\) be a finite Weyl group with generating set \(S\) and root system \(R\). Let \(X(w)\) be a Schubert variety indexed by \(w \in W\). The Schubert variety \(X(w)\) is rationally smooth if and only if the Poincaré polynomial \(P_w(q)\) is palindromic. An element \(w \in W\) is said to be rationally smooth if this condition is satisfied. Let \(V\) be the ambient Euclidean space containing \(R\). Identify \(V\) and \(V^\ast\) using the Euclidean form. Let \(R^+\) and \(R^-\) denote the positive and negative root sets of \(R\), respectively. The inversion set of \(w \in W\) is defined to be \(I(w) = \{\alpha \in R^+ : w^{-1}\alpha \in R^-\}\). The inversion hyperplane arrangement of \(w\) is \(\mathcal{I} = \bigcup_{\alpha \in I(w)} \text{ker} \, \alpha\) in \(V\). The main result of the paper is the following: Theorem. Let \(W\) be a finite Weyl group. An element \(w \in W \) is rationally smooth if and only if the inversion hyperplane arrangement \(\mathcal{I}(w)\) is free with coexponents \(d_1, \dots, d_l\) and if the product \(\prod_i(1 +d_i)\) is equal to the size of the Bruhat interval \([e, w]\). Furthermore, if \(w\) is rationally smooth then the coexponents \(d_1, \dots, d_l\) are equal to the exponents of \(w\). Schubert varieties; Coxeter groups; Weyl group; arrangements of hyperplanes; inductive freeness Slofstra, W, Rationally smooth Schubert varieties and inversion arrangements, Adv. Math., 285, 709-736, (2015) Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Rationally smooth Schubert varieties and inversion hyperplane arrangements

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The aim of this paper is threefold. Starting from previous work on monotone and strictly monotone Hurwitz numbers in terms of the fermionic Fock space, the authors derive an expression of these variants of Hurwitz numbers in terms of the bosonic Fock space. This yields to an interpretation in terms of tropical geometry involving local multiplicities given by Gromov-Witten invariants. The authors prove that a main result of Cavalieri-Johnson-Markwig-Ranganathan is actually equivalent to the Gromov-Witten/Hurwitz correspondence by Okounkov-Pandharipande for the equivariant Riemann sphere. This paper is organized as follows: The first section is an introduction to the subject. In Section 2 the authors introduce the basic notions revolving around the relation between Hurwitz numbers, Gromov-Witten theory, the Fock space and tropical geometry needed for their work. In Section 3 they derive a bosonic expression for monotone and strictly monotone Hurwitz numbers from the fermionic one. This is done by means of the boson-fermion correspondence. The fermionic expression itself is recovered from the fermionic expression for the power sums via transformations at the level of symmetric functions. The main result obtained lies in Section 4, where the authors express monotone and strictly monotone Hurwitz numbers in terms of tropical covers with local multiplicities given by Gromov-Witten invariants. This is achieved by applying Wick's theorem to the expressions obtained from the bosonification. Finally, in Section 5 the authors analyse and compare the original [\textit{A. Okounkov} and \textit{R. Pandharipande}, Ann. Math. (2) 163, No. 2, 517--560 (2006; Zbl 1105.14076)] and the tropical [\textit{R. Cavalieri} et al., Proc. Symp. Pure Math. 97, 139--167 (2018; Zbl 1451.14177)] version of Gromov-Witten/Hurwitz correpondence via the semiinfinite wedge formalism. Hurwitz numbers; Fock space; Gromov-Witten invariants; tropical geometry Enumerative problems (combinatorial problems) in algebraic geometry, Adjunction problems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Tropical jucys covers

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(W=(W,S)\) be a Coxeter group with \(S\) its distinguished generator set. Denote by \(\leqslant\) the Bruhat ordering on \(W\). \textit{D. Kazhdan} and \textit{G. Lusztig} defined polynomials \(P_{x,y}\in\mathbb{Z}[q]\) for each \(x\leqslant y\) in \(W\) [Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. These polynomials play an important role in representation theory. \textit{M. Dyer} conjectured [Compos. Math. 78, No. 2, 185-191 (1991; Zbl 0784.20019)] that \(P_{x,y}\) depends only on the isomorphism type of the poset \([x,y]=\{z\in W\mid x\leqslant z\leqslant y\}\) for the Bruhat ordering. The main result of the present paper is to give an affirmative answer to the conjecture for a class of groups, which can be stated as follows. Let \(W,W'\) be two Coxeter groups. Let \(w\in W\) and \(w'\in W'\) be such that the posets \([e,w]\) and \([e',w']\) are isomorphic for the Bruhat orderings on \(W,W'\), where \(e,e'\) are the identity elements of \(W,W'\), respectively. Then any poset isomorphism \(\phi\colon[e,w]\to[e',w']\) preserves Kazhdan-Lusztig polynomials, (in the sense that \(P_{\phi(x),\phi(y)}=P_{x,y}\) for any \(x\leqslant y\) in \([e,w]\)) in the case where one of the two groups \(W,W'\) has the property that the Coxeter graph of each of its irreducible constituents is either a tree or affine of type \(\widetilde A_n\). In particular, the result holds for all finite or affine Coxeter groups. Note that the above result was also obtained by \textit{F. Brenti} [The intersection cohomology of Schubert varieties is a combinatorial invariant, preprint (2002)] in the case where \(W,W'\) are both of type \(A_n\), as a corollary of a purely combinatorial construction of the \(P_{x,y}\). Bruhat orderings; Kazhdan-Lusztig polynomials; Schubert varieties; Coxeter groups F. du Cloux, ''Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials,'' Adv. in Math. 180 (2003), 146--175. Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Hecke algebras and their representations, Combinatorics of partially ordered sets Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This manuscript focuses on PBW degenerations of Demazure modules and Schubert varieties as interesting properties were obtained in type \(\mathrm A\), such as PBW degenerations embed naturally into the corresponding degenerate representations and flag varieties, but only with restrictions on the Weyl group element or the highest weight. I. Makhlin show that these properties cannot hold in full generality due to the issue with the definition: the degenerate variety depends on the highest weight used to define it, and not only on its Weyl group stabilizer. The author constructs minimal counterexamples for \(\mathfrak{sl}_6\) in Section \(4\), which are constructed with the aid of a study of the Cartan components appearing in this context. semisimple Lie algebras; PBW degenerations; Demazure modules; Schubert varieties Grassmannians, Schubert varieties, flag manifolds, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Combinatorial aspects of representation theory PBW degenerate Schubert varieties: Cartan components and counterexamples

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the article under review, the authors establish two goals: 1) prove a version of homological projective duality (HPD) for Pfaffian varieties as proposed by \textit{A. Kuznetsov} [Publ. Math., Inst. Hautes Étud. Sci. 105, 157--220 (2007; Zbl 1131.14017)] and 2) prove a duality between two quantum field theories called gauged linear sigma models (GLSM) as proposed by \textit{K. Hori} [J. High Energy Phys. 2013, No. 10, Paper No. 121, 74 p. (2013; Zbl 1342.81635)]. For the first result the authors build on their previous work [\textit{N. Addington} et al., Algebr. Geom. 2, No. 3, 332--364 (2015; Zbl 1322.14037)], [\textit{E. Segal} and \textit{R. Thomas}, J. Reine Angew. Math. 743, 245--259 (2018; Zbl 1454.14051)] and Kuznetzov's own work on the homological projective duality of line Grassmannians (a special case of Pfaffian varieties). The version of HPD they prove is for a non-commutative crepant resolutions (as constructed by \textit{Š. Špenko} and \textit{M. Van den Bergh} [Invent. Math. 210, No. 1, 3--67 (2017; Zbl 1375.13007)]), the reason being that in general Pfaffian varieties are very singular, as opposed to the smooth case of Grassmannians treated by Kuznetzov. The second result relies firstly on providing a mathematical, and algebro-geometric in nature, definition of B-branes for the gauged linear sigma model, which seems to be novel in the physics literature. Their definition of B-brane recovers the non-commutative crepant resolution of Pfaffian as an example of the general theory of Spenko and Van den Bergh. Finally, the duality of quantum field theories seems to be a consequence of HPD as proved in the first part. The authors concentrated in the case of an odd-dimensional vector space and the stronger results are proved in this case, as the even-dimensional case doesn't lend itself to a physical interpretation. The article has an extensive introduction that explain the main construction used in the article and also include a sketch of the proof of the main duality result. algebraic geometry; noncommutative algebra; derived categories; noncommutative resolutions Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Derived categories and associative algebras, Rings arising from noncommutative algebraic geometry Hori-mological projective duality

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper under review is concerned with the statement and the proof of a Giambelli formula holding in the integral cohomology ring \(H^*(G/P)\) of \(G/P\), where \(G\) is any classical linear algebraic group (e.g. the special linear group or any group of automorphisms of a vector space equipped with some symmetric or skew symmetric non degenerate bilinear form) and \(P\) is a parabolic subgroup of it, which means that the quotient \(G/P\) is a (homogeneous) projective variety. The homogeneous variety \(G/P\) is often called a \textsl{generalized flag variety}, because it is a natural generalization of the variety parameterizing (possibly non complete) flags of subspaces of \({\mathbb C}^n\). Among them, one should mention the orthogonal or symplectic Grassmannians or, more generally, Grassmannians of subspaces which are isotropic with respect to some non-degenerate symmetric or skew symmetric bilinear form. To explain the undoubted relevance of the result, the author spends the first few pages of the introduction to tell where does such a problem come from. The origin is the classical Schubert calculus for the usual familiar Grassmannian variety \(\mathrm{Gr}(k,n)\), which parameterizes \(k\)-dimensional vector subspaces of \({\mathbb C}^n\). The latter is of the form \(G/P\), where \(G:=\mathrm{GL}_n({\mathbb C})\) is the general linear group and \(P\) is the maximal parabolic subgroup of \(G\), obtained as the stabilizer in \(G\) of \(k\) elements of a basis of \({\mathbb C}^n\). By obvious normalizations, \(\mathrm{Gr}(k,n)\) can be also written as \(\mathrm{SL}_n({\mathbb C})/P\). It is classically known, as recalled in the introduction, that the cohomology \(H^*(\mathrm{Gr}(k,n))\) is a \({\mathbb Z}\)-module generated by the classes of the closures of the so-called Schubert cells, named Schubert cycles: the Giambelli formula expresses all Schubert cycles as explicit polynomials in certain distinguished ones, that in this case are the Chern classes of the universal quotient bundle. In the more general setting (\(G\) any classical group), the homogeneous variety \(G/P\) admits a cellular decomposition via the so-called Bruhat-cells, and its integral cohomology is generated by the cohomology classes of their closures. A first achievement of the investigation pursued in the paper under review, is the individuation of special Schubert cycles: in most cases they are, as in the usual Grassmannian issue, the Chern classes of the universal quotient bundle, but it is not so in all the cases. Using these generalized special cycles, the author is finally able to prove an analogue of Giambelli's formula for \(G/P\), where \(G\) is any classical group. What makes the job a little bit tricky is that, as it was noticed in previous literature, Giambelli formulas for non-maximal isotropic Grassmannians are not determinantal in nature (like e.g. those for the usual Grassmannians). One must substitute classical Schur polynomials by other kind of polynomials, whose definition supplied in the paper is, as it stands, a further relevant progress in the subject. Just to give a more precise flavor of the paper, the author studies the variety \(\mathfrak{X}({\mathfrak d})\) parameterizing certain partial flag of isotropic subspaces in \({\mathbb C}^{2n}\) equipped with a non degenerate skew-symmetric bilinear form. Natural Schubert varieties, indexed Weyl group of \(\mathrm{Sp}_{2n}({\mathbb C})\)), can be defined there. One of the main results expresses the classes of such Schubert varieties of \(\mathfrak{X}({\mathfrak d})\) in terms of the substitutes of Schubert polynomials alluded to above. This central result, which is in fact a Giambelli formula for the considered situation, is not just isolated but amazingly embedded in a paper that is rich of intriguing propositions, which are interesting in their own. It is worth to add that the nice Section 1, devoted to the preliminaries, is very helpful and is a concrete attempt to keep the paper fairly self-contained. Another nice feature, which enlarges the perspective of the results, is that the aforementioned main theorem extends also to the case of torus-equivariant cohomology ring \(G/P\) and to the setting of symplectic and orthogonal degeneracy loci. The paper ends with a huge and probably complete reference list, which is very useful not just for the interested reader, but fundamental for all researchers aiming to draw their own path for studying and possibly working on this hot subject. homogeneous varieties; classical and equivariant Schubert Calculus; Giambelli formula; Schubert polynomials; bi-Tableaux; mixed Stanley functions Tamvakis, H., \textit{A Giambelli formula for classical G/P spaces}, J. Algebraic Geom., 23, 245-278, (2014) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus A Giambelli formula for classical \(G/P\) spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\), \(Y\) be finite sets and \(T\) a set of functions \(X \rightarrow Y\) which we will call ``tableaux''. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch's set-valued semistandard tableaux. One vertex decomposition of this ``Young tableau complex'' parallels Lascoux's transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials. Young tableaux; simplicial complex; semistandard Young tableaux; Lascoux transition formula A. Knutson, E. Miller, and A. Yong. ''Tableau complexes''. Isr. J. Math. 163.1 (2008), pp. 317-- 343.DOI. 12 Cara Monical Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds Tableau complexes

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors explore the relation between Schubert calculus and volume polynomial on spaces of convex polytopes. They give representations of Schubert cycles in a complete flag manifold by sums of faces of the Gelfand-Tsetlin polytope. They are motivated by interplay between algebraic geometry and convex poytopes, originally explored for toric varieties. Their main tool is a construction which associates with each convex polytope \(P\) in \(\mathbb R^d\) a graded commutative ring \(R_P\), called polytope ring, satisfying Poincaré duality. Here the polytope ring \(R_P\) is isomorphic to the Chow ring of the corresponding smooth toric variety \(X_P\). Faces of \(P\) give elements of \(R_P\), which generate \(R_P\) as an abelian group. If \([F]\) is the element of \(R_P\) corresponding to a face \(F\), then \([F] \dot [G] = [F \cap G]\) in \(R_P\), provided that they are transverse. Individual faces of \(P\) represent cycles given by the closures of the torus orbits in \(X_P\). In this paper they arwe interested in the case when \(P\) is not simple. They are using the relation between Gelfand-Tsetlin polytopes associated with dominant weight of the group \(\mathrm{GL}_n(\mathbb C)\) and Chow ring of variety \(X\) of complete flags in \(\mathbb C^n\). One of their results is a general construction that associates with each element of \(R_P\) a linear combination of faces of \(P\). They embed the ring \(R_P\) in a \(\mathbb Z\) module \(M_P\) whose elements can be regarded as linear combinations of arbitrary faces of \(P\) modulo some relations. The module \(M_P\) depends on the choice of a resolution of \(P\). On the algebro-geometric level \(R_P\) can be regarded as the subring of Chow ring of the singular toric variety \(X_P\) generated by the Picard group, and \(M_P\) can be constructed using a resolution of singularities for \(X_P\). If this construction is applied to the Gelfand-Tsetlin polytope, Chow ring of flag variety \(X\) is afree an abelian group with a basis of Schubert cycles indexing by permutations. So here the Schubert cycles represent linear combinations of faces of the Gelfan-Tsetlin polytope. So this representation has applications in Schubert calculus. The authors describe cup product of Schubert cycles by reduced Kogan and dual Kogan faces. Also they give Demazure Character formula by reduced Kogan faces with permutations in the Gelfand-Tsetlin polytope associated with weight \(\lambda\). The paper is nice and it is well written. Gelfand-Tsetlin polytope; Schubert calculus; Demazure Character formula; Kogan faces; Chow ring; flag varieties; volume polynomial Kirichenko, V. A.; Smirnov, E. Yu.; Timorin, V. A., Schubert calculus and Gelfand-zetlin polytopes, Uspekhi Mat. Nauk, 67, 89-128, (2012) Group actions on varieties or schemes (quotients), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Schubert calculus and Gelfand-Zetlin polytopes

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we investigate properties of modules introduced by \textit{W. Kraśkiewicz} and \textit{P. Pragacz} [C. R. Acad. Sci., Paris, Sér. I 304, 209--211 (1987; Zbl 0642.13011); Eur. J. Comb. 25, No. 8, 1327--1344 (2004; Zbl 1062.14065)] which realize Schubert polynomials as their characters. In particular, we give some characterizations of modules having filtrations by Kraśkiewicz-Pragacz modules. In finding criteria for such filtrations, we calculate generating sets for the annihilator ideals of the lowest vectors in Kraśkiewicz-Pragacz modules and derive a projectivity result concerning Kraśkiewicz-Pragacz modules. Schubert polynomials; Kraśkiewicz-Pragacz modules Watanabe, M.: An approach toward Schubert positivities of polynomials using kraśkiewicz-pragacz modules. European J. Combin. 58, 17-33 (2016) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds An approach towards Schubert positivities of polynomials using Kraśkiewicz-Pragacz modules

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(V\) be a symplectic vector space and \(LG\) the Lagrangian Grassmannian which parameterizes the maximal isotropic subspaces in \(V\). In the paper under review the author studies the quantum cohomology ring \(QH^*(LG)\) and proves that its multiplicative structure is determined by the ring of \(Q\)-polynomials. quantum cohomology; Lagrangian Grassmannian; \(Q\)-polynomials. Kresch A., Tamvakis H.: Quantum cohomology of the Lagrangian Grassmannian. J. Algebraic Geom. 12, 777--810 (2003) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of the Lagrangian Grassmannian.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author describes the irreducible components of the singular locus of Schubert varieties in flag manifolds. The full flag variety of flags in a vector space of dimension \(n\) has a cellular decomposition corresponding to Schubert varieties \(X_w\) indexed by the permutations \(\mathfrak S_n\) of \(1,\dots, n\). For a fixed flag \(V_1 \subset \cdots \subset V_n\) the Schubert variety \(X_w\) is given by the flags \(W_1\subset \cdots \subset W_n\) such that \(\dim(W_p\cap V_q)\geq r_w(p,q)\) for \(1\leq p,q\leq n\), where \(r_w(p,q) =\#\{i\leq p, w(i)\leq q\}\). By a result by \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci. Math. Sci. 100, 45--52 (1990; Zbl 0714.14033)] the variety \(X_w\) is smooth if and only if there is no quadruple of integers \(i<j<k<l\) such that there are configurations \(w(l)< w(j)< w(k)< w(i)\) or \(w(k) <w(l)< w(i) <w(j)\). The author describes an explicit process that from minimal configurations of the above type produces a set \(C(w)\) such that the irreducible components of the singular locus of \(X_w\) are exactly the Schubert varieties \(X_v\) such that \(v\in C(w)\). From this result the author can describe the irreducible components of the Schubert varieties in manifolds of not necessarily complete flags. The results have been obtained independently by \textit{S. C. Billey} and \textit{G. S. Warrington} [Trans. Am. Math. Soc. 355, 3915--3945 (2003; Zbl 1037.14020)], and by \textit{C. Kassel, A. Lascoux}, and \textit{C. Reutenauer} [Adv. Math. 150, 1--35 (2000; Zbl 0981.15009)]. flag varieties; singular locus of Schubert varieties; irreducible components Manivel, L., Le lieu singulier des variétés de Schubert, Int. Math. Res. Not., 16, 849-871, (2001), MR 1853139 Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Determinantal varieties The singular locus of Schubert varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The result of the author, \textit{A. Kresch} and \textit{H. Tamvakis} [J. Am. Math. Soc. 16, No. 4, 901--915 (2003; Zbl 1063.53090)] shows that the computation of the (3- point, genus 0) Gromov-Witten invariants of Grassmann variety can be reduced to a computation in the ordinary cohomology of certain two-step flag manifolds. The Gromov-Witten invariants, in this case, have a nice enumerative interpretation. Namely, they count rational curves that meet general translates of Schubert varieties. Unfortunately, the Gromov-Witten invariants used to define more general quantum cohomology theories do not have such an interpretation. Despite this, the authors prove that the more general Gromov-Witten invariants satisfy the key identity from the paper of Buch, Kresch, and Tamvakis. One of the main results of the paper is that an equivariant \(K\)-theoretic Gromov-Witten invariant on a Grassmannian is equal to a quantity computed in the ordinary equivariant \(K\)-theory of a two-step flag variety. In the process of proving this the authors show that the Gromov-Witten variety of curves passing through three general points is rational and irreducible. They also describe the structure of the quantum \(K\)-theory ring of Grassmanian in terms of a Pieri rule and compute the dual Schubert basis for this ring. The authors show that the formula for Gromov-Witten invariants can be partially generalized to cominuscule homogeneous spaces. This is proven by using a construction of \textit{P.-E. Chaput, L. Manivel} and \textit{N. Perrin} [Transform. Groups 13, No. 1, 47--89 (2008; Zbl 1147.14023)]. Gromov-Witten invariants; quantum \(K\)-tkeory; Grasssmannian Buch, A. S.; Mihalcea, L. C., Quantum \textit{K}-theory of Grassmannians, Duke Math. J., 156, 3, 501-538, (2011) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), \(K\)-theory of schemes, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Rationality questions in algebraic geometry Quantum \(K\)-theory of Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we explore a consequence of symplectic duality (also known as 3d mirror symmetry) in the setting of enumerative geometry. The theory of quasimaps allows one to associate hypergeometric functions called vertex functions to quiver varieties. In this paper, we prove a formula which relates the vertex functions of \(T^\ast Gr(k,n)\) and its symplectic dual. In the course of the proof, we study a family of \(q\)-difference operators which act diagonally on Macdonald polynomials. Our results may be interpreted from a combinatorial perspective as providing an evaluation formula for a \(q\)-Selberg type integral. Symplectic duality for \(T^\ast Gr(k,n)\)

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Any dualizable object \(X\) in a symmetric monoidal category \((\mathcal{C},\otimes,\mathbf{1})\) gives rise to a categorical Euler characteristic \[ \mathbf{1}\to X\otimes D(X)\to D(X)\otimes X\to \mathbf{1} \] which is an endomorphism of the unit object. In the case where \(\mathcal{C}\) is the stable homotopy category, with the smash product as symmetric monoidal structure, the endomorphisms of the sphere spectrum form the ring of integers, and this categorical construction recovers the classical Euler characteristic of finite polyhedra. The case of the Morel-Voevodsky stable homotopy category \(SH(k)\) of a field \(k\) recently received substantial attention. Morel proved that the endomorphisms of the sphere spectrum \(\mathbf{1}_k\) over \(k\) form the Grothendieck-Witt ring \(\mathrm{GW}(k)\) of nondegenerate symmetric bilinear forms over \(k\). Work of Voevodsky and Ayoub implies that every smooth projective variety \(X\) over \(k\) is dualizable in \(SH(k)\), and hence possesses the class of a nondegenerate symmetric bilinear form as its motivic or \(\mathbb{A}^1\)-Euler characteristic. In the article under review, the authors provide an explicit model for the \(\mathbb{A}^1\)-Euler characteristic of a smooth projective variety \(X\) over a field of characteristic zero: It is the Hochschild homology \(\mathrm{HH}(X)\), equipped with a canonical symmetric pairing \(B_X\) on \(\mathrm{HH}(X)\) which is nondegenerate by \textit{L. Alonso Tarrío} et al. [Adv. Math. 257, 365--461 (2014; Zbl 1301.14007)]. Their proof uses among other ingredients the Hochschild-Kostant-Rosenberg Theorem, as well as a different model for the Euler characteristic provided by \textit{M. Levine} and \textit{A. Raksit} [Algebra Number Theory 14, No. 7, 1801--1851 (2020; Zbl 1458.14029)]. Theorem 2.13 identifies their construction as a motivic measure, that is, a ring homomorphism from the Grothendieck ring of varieties over \(k\) to the Grothendieck-Witt ring over \(k\). The final section illustrates the applicability of these explicit models for computations in the cases of \(\mathbb{P}^n\) for \(n\in \{1,2\}\) and the blow-up of a point in the projective plane. \(\mathbb{A}^1\)-Euler characteristic; Grothendieck-Witt group; Hochschild cohomology; Hermitian \(K\)-theory Motivic cohomology; motivic homotopy theory, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) Compactly supported \(\mathbb{A}^1\)-Euler characteristic and the Hochschild complex

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [Part I, cf. Invent. Math. 79, 499-511 (1985; Zbl 0563.14023).] We want to develop a theory of polynomials for the parabolic setup for any Coxeter group (W,S) and the subgroup \(W_ J\) generated by any subset \(J\subseteq S\). This is the starting point of our investigation. It turns out that one does get a set \(\{P^ J_{\tau,\sigma}\}\) of polynomials in \({\mathbb{Z}}[q]\) which is indexed by a pair \(\tau\), \(\sigma\) of elements in \(W^ J\), the set of minimal coset representatives of \(W/W_ J\). These polynomials give the dimensions of the intersection cohomology modules of Schubert varieties in G/P (see Theorem 4.1) for any P. They are related to \(P_{x,y}'s\) when the subgroup corresponding to P is finite (see Propositions 3.4 and 3.5). Incidentally, Proposition 3.5 provides a method for computing \(P_{x,y}'s\) which is very efficient since the number of intermediate steps is considerably smaller than that in the original setup (for any Coxeter group (W,S)). Kazhdan Lusztig polynomials; Verma modules; semisimple algebraic group; Kac-Moody groups; Coxeter group; coset representatives; intersection cohomology modules; Schubert varieties V.V. Deodhar, \textit{On some geometric aspects of Bruhat orderings. II. The parabolic analogue of Kazhdan-Lusztig polynomials}, \textit{J. Algebra}\textbf{111} (1987) 483. Infinite-dimensional Lie groups and their Lie algebras: general properties, Cohomology theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds, Semisimple Lie groups and their representations, Representation theory for linear algebraic groups, Other algebraic groups (geometric aspects) On some geometric aspects of Bruhat orderings. II: The parabolic analogue of Kazhdan-Lusztig polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define the \textit{double Gromov-Witten invariants} of Hirzebruch surfaces in analogy with double Hurwitz numbers, and we prove that they satisfy a piecewise polynomiality property analogous to their one-dimensional counterpart. Furthermore we show that each polynomial piece is either even or odd, and we compute its degree. Our methods combine floor diagrams and Ehrhart theory. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Coverings of curves, fundamental group The double Gromov-Witten invariants of Hirzebruch surfaces are piecewise polynomial

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the torus equivariant Schubert classes of the Grassmannian of non-maximal isotropic subspaces in a symplectic vector space. We prove a formula that expresses each of those classes as a \textit{sum} of multi Schur-Pfaffians, whose entries are equivariantly modified special Schubert classes. Our result gives a proof to Wilson's conjectural formula, which generalizes the Giambelli formula for the ordinary cohomology proved by Buch-Kresch-Tamvakis, given in terms of Young's raising operators. Furthermore we show that the formula extends to a certain family of Schubert classes of the symplectic partial isotropic flag varieties. Schubert classes; symplectic Grassmannians; torus equivariant cohomology; Giambelli type formula; Wilson's conjecture; double Schubert polynomials Ikeda, T.; Matsumura, T., \textit{Pfaffian sum formula for the symplectic Grassmannians}, Math. Z., 280, 269-306, (2015) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pfaffian sum formula for the symplectic Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Our purpose is to describe some remarks on Schur polynomials, which play an important role in the theory of Chow ring. In {\S} 2, we describe the relationship between the Schur polynomials and the Schubert cycles of a Grassmann variety. In {\S} 3, we prove the duality of Schur polynomials using the theory of the Chow ring of Grassmann variety. - In {\S} 4, we present the theorem of W. Fulton and R. Lazarsfeld for numerically positive polynomials for ample vector bundles. - In {\S} 5 and {\S} 6, we introduce the Gysin's projection formula for flag bundles and give a formal proof of this formula. Schur polynomials; Schubert cycles; Chow ring of Grassmann variety; flag bundles Grassmannians, Schubert varieties, flag manifolds, Parametrization (Chow and Hilbert schemes) Remarks on Schur polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article is a sequel to the earlier work [\textit{M. Marcolli} and \textit{G. Tabuada}, Sel. Math., New Ser. 20, No. 1, 315--358 (2014; Zbl 1296.14002)]. The author develops a notion of noncommutative mixed motives with coefficients in a commutative ring, and explains how their construction is compatible with commutative mixed motives (Theorem 2.1). He introduces noncommutative mixed Artin motives, and he proves that the category of noncommutative mixed Artin motives associated to a characteristic 0 field does not coincide with its classical category of mixed Artin motives (Proposition 2.4). The author applies work of Ayoub to construct a motivic Hopf dg algebra associated to their theory of non-commutative mixed motives (Proposition 2.5). The motivic Hopf dg algebra is the mixed analogue of the motivic Galois group in the setting of noncommutative numerical motives (see the earlier paper [\textit{M. Marcolli} and \textit{G. Tabuada}, J. Eur. Math. Soc. (JEMS) 18, No. 3, 623--655 (2016; Zbl 1376.14004)]). The author points out that, while the existence of the afore-mentioned motivic Galois group requires assuming noncommutative analogues of standard conjectures C and D, the motivic Hopf dg algebra always exists. The author goes on to establish several properties of the motivic Hopf algebra (e.g. Theorem 2.10) which mirror properties of the motivic Galois group. Hopf dg algebra; weak Tannakian formalism; Hochschild homology; algebraic \(K\)-theory; mixed Artin-Tate motives; orbit category; noncommutative algebraic geometry ______, {\em Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras}. Selecta Mathematica, \textbf{22} (2016), no.~2, 735--764 Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.), Hopf algebras and their applications, \(K\)-theory and homology; cyclic homology and cohomology Noncommutative mixed (Artin) motives and their motivic Hopf dg algebras

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The discovery of classical and quantum completely integrable systems led to an increase of interest in the theory of Abelian functions in theoretical physics and applied mathematics. In this paper the authors consider an arbitrary algebraic curve \(V\) of genus \(g\) and construct the field of meromorphic functions on its Jacobi variety \(\text{Jac}(V)\) in terms of Kleinian \(\rho\)-functions \[ \rho_{ij}(u) = - \frac{\partial^{2}}{\partial u_{i}\partial u_{j}}\ln \sigma (u), \qquad i, j = 1, 2,\dots , g, \] where the vector \(u \in \text{Jac}(V)\) and \(\sigma\) is the Kleinian \(\sigma\)-function. The effective construction of the \(\sigma\)-function plays the principal role in the authors' approach. It is defined on the universal space of Jacobians, which is the fibration with the base given by the moduli space \(M(V)\) of the curve \(V\) of dimension \(d \leq 3g - 3\) and a fibre generated by the Jacobi variety \(\text{Jac}(V).\) The Kleinian \(\sigma\) function represents a natural generalization of the Weierstrass elliptic function to the case of an arbitrary algebraic curve. The principal result of the paper is the explicit solution of the Jacobi inversion problem, which is an alternative to that given by M. Noether. The paper is completed by a short discussion on the application of the authors' approach to completely integrable equations and of further perspectives of development of the theory. quantum completely integrable systems; Jacobi variety; Jacobi inversion problem J. C. Eilbeck, V. Z. Enolskii, and D. V. Leykin, ''On the Kleinian Construction of Abelian Functions of Canonical Algebraic Curves,'' in SIDE III: Symmetries and Integrability of Difference Equations: Proc. Conf., Sabaudia, Italy, 1998 (Am. Math. Soc., Providence, RI, 2000), CRM Proc. Lect. Notes 25, pp. 121--138. Jacobians, Prym varieties, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Riemann surfaces; Weierstrass points; gap sequences, KdV equations (Korteweg-de Vries equations) On the Kleinian construction of Abelian functions of canonical algebraic curves

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In her thesis, \textit{M. Walters} [Geometry and uniqueness of some extreme subvarieties in complex Grassmannians, Ph.D. thesis, University of Michigan (1997)] studied subvarieties \(X\) of the complex Grassmannian with the property that \([X]=r[X_w]\), \(r\in {\mathbb Z}\) , where \(X_w\) is a Schubert variety of type \(w\) and [ ] denotes homology class. She does this by introducing two differential systems \({\mathcal R}_w\) and \({\mathcal B}_w\), called the Schur differential system and the Schubert differential system respectively, and discussing some related rigidity questions. Schur and Schubert rigidity of several kinds of smooth and singular Schubert varieties in Hermitian symmetric space were investigated systematically by \textit{R. Bryant} [Rigidity and quasi-rigidity of extremal cycles in compact Hermitian symmetric spaces. Ann. Math. Stud. 153 (2007; Zbl 01948238)]. He proves, among other things, that \(X_w\) is Schur rigid when it is either a maximal linear space in a Hermitian symmetric space or is a sub-Lagrangian Grassmannian in a Lagrangian Grassmanian. In the paper under review, the author generalizes this result, proving that any smooth Schubert variety in a Hermitian symmetric space, other than an odd-dimensional quadric, is Schur rigid except when it is a non-maximal linear space. Showing \(X_w\) is Schur rigid is equivalent to showing that \({\mathcal R}_w = {\mathcal B}_w\) and \(X_w\) is Schubert rigid. Using a result of Goncharov, the author shows that \(X_w\) will be Schubert rigid if the \((1,1)\) cohomology group of the fiber of \({\mathcal B}_w\) is zero. The vanishing of this cohomology group and the equality of the two differential systems are established for smooth Schubert varieties by the use of Dynkin diagrams and Lie algebra cohomology. Hong J.: Rigidity of smooth Schubert varieties in Hermitian symmetric spaces. Trans. Am. Math. Soc. 359(5), 2361--2381 (2007) (electronic) Algebraic cycles, Grassmannians, Schubert varieties, flag manifolds, Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study Thom polynomials of singularities. For a (so called right-left) complex singularity \(\eta\), and a map \(f\) between compact complex manifolds, one can consider the singularity subset as the collection of those points in the source where the map has singularity \(\eta\). It is known that the closure of this set (under favorable circumstances) carries a cohomology class, which can be calculated by substituting the characteristic classes of the source and target manifold (pulled back via \(f\)) into a universal polynomial, the Thom polynomial of the singularity. The authors study the positivity properties of Thom polynomials. More generally they study the positivity properties of \(G\)-equivariant cohomology classes represented by invariant cones in representations of \(G\), where \(G\) is a product of general linear groups. They prove that if the representation is determined by a functor which preserves ``global generatedness'', then these equivariant classes, when expressed in the basis of products of Schur functions of the Chern roots of the general linear groups, have \textit{non-negative} coefficients. The proof eventually reduces this statement to an appropriate positivity result of Lazarsfeld and Fulton. As an application, the authors show that the Thom polynomials of right-left singularities have non-negative coefficients when expressed in the basis of products of Schur functions of the source and target Chern roots. This result extends earlier results of the authors concerning less general singularity types. Thom polynomials; Schur functions Pragacz, P., Weber, A.: Thom polynomials of invariant cones, Schur functions and positivity. In: Algebraic cycles, sheaves, shtukas, and moduli, Trends Math., pp. 117-129. Birkhäuser, Basel (2008) Classical problems, Schubert calculus, Symmetric functions and generalizations, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Homology of classifying spaces and characteristic classes in algebraic topology Thom polynomials of invariant cones, Schur functions and positivity

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G_0\) be a real semisimple Lie group. It acts naturally on every complex flag manifold \(z= G/Q\) of its complexification. Given an Iwasawa decomposition \(G_0= K_0 A_0 N_0\), a \(G_0\)-orbit \(\gamma\subset Z\), and the dual \(K\)-orbit \(\kappa \subset Z\), Schubert varieties are studied and a theory of Schubert slices for arbitrary \(G_0\)-orbits is developed. For this, certain geometric properties of dual pairs \((\gamma,\kappa)\) are underlined. Canonical complex analytic slices contained in a given \(G_0\)-orbit which are transversal to the dual \(K_0\)-orbit \(\gamma\cap \kappa\) are constructed and analyzed. Associated algebraic incidence divisors are used to study complex analytic properties of certain cycle domains. In particular, it is shown that the linear cycle space \(\Omega_W(D)\) is a Stein domain that contains the universally defined Iwasawa domain \(\Omega_I\). This is one of the main ingredients in the proof that \(\Omega_W(D)= \Omega_{AG}\) for all but a few Hermitian exceptions. In the Hermitian case, \(\Omega_W(D)\) is concretely described in terms of the associated bounded symmetric domain. real semisimple Lie group; flag manifold; incidence division Huckleberry A., Wolf J.A. (2003): Schubert varieties and cycle spaces. Duke Math. J. 120, 229--249,(AG/0204033) Stein spaces, Analysis on real and complex Lie groups, Grassmannians, Schubert varieties, flag manifolds, Homogeneous complex manifolds Schubert varieties and cycle spaces

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Counting the number of algebraic curves and varieties subject to various conditions is one basic problem in the enumerative algebraic geometry, and Schubert calculus is a systematic and effective theory to solve such problems. It was developed by Schubert, and his most comprehensive and accessible exposition of this theory is given in this book. Right from the beginning, the theory of Schubert calculus has attracted the attention of many great mathematicians. For example, Hilbert proposed a rigorous justification of Schubert calculus as the 15th problem in his famous list of 23 problems. Recent developments in string theory have contributed to solutions of some outstanding problems in enumerative geometry, and, hence, greatly renewed interest in this subject. The English translation of this classic by Schubert will be most valuable and interesting to both beginners and experts in enumerative geometry in order to learn how Schubert thought about the problems and how he proposed to solve them, in particular to appreciate the freshness of the subject under development. As Schubert put it: this book ``should acquaint the reader with the ideas, problems and results of a new area of geometry'' and ``should teach the handling of a peculiar calculus that enables one to determine in an easy and natural way a great many of those geometric numbers and relations between singularity numbers.'' See the review of the 1979 reprint edition of the 1879 original in [Zbl 0417.51008]. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry, Real and complex geometry, Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry, Classical problems, Schubert calculus, Collected or selected works; reprintings or translations of classics, History of geometry The calculus of enumerative geometry. Translated from the German by Wolfgang Globke

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{P. Edelman} and \textit{C. Greene} [Adv. Math. 63, 42--99 (1987; Zbl 0616.05005)] generalized the Robinson-Schensted-Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman-Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well. Schubert polynomials; Demazure characters; key polynomials; RSK; Edelman-Greene insertion; reduced words Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Combinatorial aspects of algebraic geometry, Symmetric functions and generalizations, Permutations, words, matrices, Exact enumeration problems, generating functions, Combinatorial identities, bijective combinatorics A generalization of Edelman-Greene insertion for Schubert polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of \({\mathbb{Z}}\) generated by the class of a point. This clarifies the extent to which the graph hypersurfaces `generate the Grothendieck ring of varieties': while it is known that graph hypersurfaces generate the Grothendieck ring over a localization of \({\mathbb{Z}[\mathbb{L}]}\) in which \({\mathbb{L}}\) becomes invertible, the span of the graph hypersurfaces in the Grothendieck ring itself is nearly killed by setting the Lefschetz motive \({\mathbb{L}}\) to zero. In particular, this shows that the graph hypersurfaces do \(not\) generate the Grothendieck ring prior to localization. The same result yields some information on the mixed Hodge structures of graph hypersurfaces, in the form of a constraint on the terms in their Deligne-Hodge polynomials. These observations are certainly not surprising for the expert reader, but are somewhat hidden in the literature. The treatment in this note is straightforward and self-contained. graph hypersurfaces; Grothendieck ring; stable birational equivalence Aluffi P., Marcolli M.: Graph hypersurfaces and a dichotomy in the Grothendieck ring. Lett. Math. Phys. 95, 223--232 (2011) (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Perturbative methods of renormalization applied to problems in quantum field theory, Rationality questions in algebraic geometry, Grothendieck groups, \(K\)-theory, etc. Graph hypersurfaces and a dichotomy in the Grothendieck ring

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{A. Skovsted Buch} and \textit{W. Fulton} [Invent. Math. 135, No. 3, 665--687 (1999; Zbl 0942.14027)] gave a formula for a general kind of degeneracy locus associated to an oriented Type A quiver. This formula involves Schur determinants and the quiver coefficients which generalize the classical Littlewood-Richardson coefficients. In the paper under review, the authors give a positive combinatorial formula for the quiver coefficients when the rank conditions defining the degeneracy locus are given by a permutation. As applications, one obtains new expansions for Fulton's universal Schubert polynomials, Schubert polynomials of Lascoux-Schützenberger, and explicit Giambelli formulas in the classical and quantum cohomology ring of any partial flag variety. quiver coefficients \beginbarticle \bauthor\binitsA. S. \bsnmBuch, \bauthor\binitsA. \bsnmKresch, \bauthor\binitsH. \bsnmTamvakis and \bauthor\binitsA. \bsnmYong, \batitleSchubert polynomials and quiver formulas, \bjtitleDuke Math. J. \bvolume122 (\byear2004), no. \bissue1, page 125-\blpage143. \endbarticle \OrigBibText Anders S. Buch, Andrew Kresch, Harry Tamvakis, and Alexander Yong, Schubert polynomials and quiver formulas , Duke Math. J. 122 (2004), no. 1, 125-143. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Schubert polynomials and quiver formulas

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac-Moody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construct large families of quantum clusters for all of these algebras and the quantum Richardson varieties associated to arbitrary symmetrizable Kac-Moody algebras and all pairs of Weyl group elements. Along the way we develop a quantum version of the Fomin-Zelevinsky twist map for all quantum Richardson varieties. Furthermore, we establish an explicit relationship between the Goodearl-Letzter and Cauchon approaches to the descriptions of the spectra of symmetric CGL extensions. Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Kac-Moody groups, Simple and semisimple modules, primitive rings and ideals in associative algebras, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected, simply connected and simple algebraic group over \(\mathbb{C}\). Fix a Borel subgroup \(B\), its maximal torus \(H\) and the Weyl group \(W\). Let \(P^+\) denote the set of integral dominant weights. Consider the Grothendieck ring \(K_H(G/B)\) of the category of \(H\)-equivariant coherent sheaves on the flag variety \(G/B\). It has a basis over the Laurent polynomial ring \(K_H(\mathrm{pt})\) given by the the classes \([\mathcal{O}_{X_w}]\) of the structure sheaves of Schubert varieties \(X_w\) for \(w \in W\). The Pieri-Chevalley formula expresses the product \([\mathcal{L}_{\lambda}] [\mathcal{O}_{X_w}]\) in this basis; here \(\lambda \in P^+\) and \(\mathcal{L}_{\lambda} \longrightarrow G/B\) is the line bundle arising from the one-dimensional \(B\)-module of weight \(\lambda\). \textit{P. Littelmann} and \textit{C. S. Seshadri} [Prog. Math. 210, 155--176 (2003; Zbl 1100.14526)] related the above linear combination to the standard monomial theory for finite-dimensional \(G\)-modules. In more details, for \(\lambda \in P^+\), the simple \(G\)-module \(L(\lambda)\) of highest weight \(\lambda\) has a basis \((p_{\pi})\) indexed by Littelmann-Seshadri paths \(\pi\) of shape \(\lambda\). Let \(\mu \in P^+\) and view \(L(\lambda+\mu)\) as a submodule of the tensor product \(L(\lambda) \otimes L(\mu)\). The standard monomial theory provides a monomial basis \((p_{\pi} p_{\eta})\) of \(L(\lambda+\mu)\), indexed by pairs of Littelmann-Seshadri paths \(\pi\) and \(\eta\) of shapes \(\lambda\) and \(\mu\) respectively, satisfying a certain standard property. Then the linear combination for \([\mathcal{L}_{\lambda}][\mathcal{O}_{X_w}]\) is encoded in the monomial bases of \(V(\lambda+\mu)\) for specific choices of \(\mu\). Let \(U_q(\mathfrak{g}_{\mathrm{aff}})\) denote the quantum affine algebra associated to the affinization \(\mathfrak{g}_{\mathrm{aff}}\) of the Lie algebra of \(G\), and \(W_{\mathrm{aff}}\) the affine Weyl group, which is a semi-direct product of \(W\) with the integral coweight lattice of \(G\). For \(\lambda \in P^+\) and \(x \in W_{\mathrm{aff}}\), Kashiwara introduced the level zero extremal module \(V(\lambda)\) over \(U_q(\mathfrak{g}_{\mathrm{aff}})\) and its Demazure submodule \(V_x(\lambda)\) over the nilpotent subalgebra \(U_q^-(\mathfrak{g}_{\mathrm{aff}})\), and equipped both modules with compatible crystal structure. The recent works [\textit{M. Ishii} et al., Adv. Math. 290, 967--1009 (2016; Zbl 1387.17028)] and [\textit{S. Naito} and \textit{D. Sagaki}, Math. Z. 283, No. 3--4, 937--978 (2016; Zbl 1395.17029)] described the crystal structure in terms of semi-infinite Littelmann-Seshadri paths. In the present work the authors generalize the Pieri-Chevalley formula to an ``affine version'' of \(G/B\), the semi-infinite flag manifold \(\mathbf{Q}_G\). For that purpose, the authors establish fundamental results on semi-infinite Schubert varieties \(\mathbf{Q}_G(x)\) indexed by \(x \in W_{\mathrm{aff}}\) (actually one needs to replace the integral coweight lattice by the cone). Notably, for \(\lambda \in P^+\), the structure sheaf \(\mathcal{O}_{\mathbf{Q}_G(x)}\) twisted by the line bundle \(\mathcal{L}_{\lambda}\) has vanishing higher cohomology and its space of global sections is identified with the Demazure submodule \(V_x(-w_0 \lambda)\), where \(w_0\) denotes the longest element in the Weyl group \(W\). These results enable the authors to have a good definition of K-theory of \(\mathbf{Q}_G\), equivariant with respect to the Iwarahori subgroup of \(G(\mathbb{C}[[z]])\). To express \([\mathcal{L}_{\lambda}][\mathcal{O}_{\mathbf{Q}_G(x)}]\) as a linear combination of the classes \([\mathcal{O}_{\mathbf{Q}_G(y)}]\) for \(y \in W_{\mathrm{aff}}\), as in the classical case of Littelmann-Seshadri, the authors develop a standard monomial theory for crystal bases of Demazure modules. semi-infinite flag manifold; normality; \(K\)-theory; Pieri-Chevalley formula; standard monomial theory; semi-infinite Lakshmibai-Seshadri path Quantum groups (quantized enveloping algebras) and related deformations, Classical problems, Schubert calculus, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations Equivariant \(K\)-theory of semi-infinite flag manifolds and the Pieri-Chevalley formula

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is a classical result in representation theory that to each partition \(\lambda\) one may attach a finite-dimensional representation \(V(\lambda)\) of the general linear group \(\mathrm{GL}_r({\mathbb C})\). The tensor product of any two such representations decomposes into the direct sum of irreducible representations whose multiplicities are precisely those prescribed by the Littlewood-Richardson rule. This fact implies a non-trivial and beautiful relationship with Schubert Calculus for the complex Grassmannians \(\mathrm{Gr}(r,n)\), because the structural constants of the singular cohomology \(H^*(\mathrm{Gr}(r,n),{\mathbb Q})\) of \(\mathrm{Gr}(r,n)\) are precisely the Littlewood-Richardson coefficients. This occurrence can be expressed by saying that there is an explicit surjective homomorphism \[ \xi: \mathrm{Rep}_{\mathrm{poly}}^{{\mathbb C}}(L)\rightarrow H^*(\mathrm{Gr}(r,n)),\leqno{(*)} \] where \(\mathrm{Rep}^{\mathbb C}_{\mathrm{poly}}\) denotes the complexification of the polynomial representation ring of \(\mathrm{GL}_r\), mapping \(V(\lambda)\) to \(\sigma_\lambda\), the Schubert cycle associated to the partition \(\lambda\). For more on this relationship, from a different point of view, the nice survey by \textit{H. Tamvakis} [Enseign. Math. (2) 50, No. 3--4, 267--286 (2004; Zbl 1131.14311)] is recommended. The purpose of the paper under review is to explore how much such a phenomenology can be generalized for other flag varieties of the form \(G/P\), by replacing the general linear group \(\mathrm{GL}_r({\mathbb C})\) with any Levi subgroup \(L\) of any reductive group \(G\) and the corresponding generalized flag variety. More precisely, one starts from the datum \((G,B, P, L)\), where \(G\) is a connected reductive group \(G\) over \({\mathbb C}\), \(B\) a Borel subgroup of it with maximal torus \(T\), \(P\) a parabolic subgroup and \(L\) a Levi subgroup of \(P\) containing \(T\). Then, the role of the partitions in the map \((*)\) is now played by the set \(D\) of the dominant characters of \(T\), so that the main achievement of the paper, summarized in Theorem 5, consists in the construction of a surjective \({\mathbb C}\)-algebra homomorphism \[ \mathrm{Rep}_{\lambda-\mathrm{poly}}^{{\mathbb C}}(L)\rightarrow H^*(G/P, {\mathbb C})\leqno{(**)} \] where \(\mathrm{Rep}_{\lambda-\mathrm{poly}}^{{\mathbb C}}(L)\) is the \(\lambda\)-polynomial subring of \(\mathrm{Rep}^{\mathbb C}(L)\), the complexification of the representation ring \(\mathrm{Rep} (L)\) of \(L\), which maps any almost faithful irreducible \(G\)-module \(V(\lambda)\) to a cohomology class of the flag variety \(G/P\). The reviewer is aware that many of the terms used in the review are not explained. To keep it to a reasonable length the non expert but interested reader is necessarily invited to look at the paper for the background. The \(\lambda\)-polynomial subring occurring in the map \((**)\), and the map itself, is defined via a morphism \(\theta_\lambda:G\rightarrow {\mathfrak g}\), where \({\mathfrak g}\) is the Lie algebra of \(G\), known in the literature as Springer morphism, well described in the paper under review or in the original source [\textit{T. A. Springer}, in: Algebr. Geom., Bombay Colloq. 1968, 373--391 (1969; Zbl 0195.50803)], or in Section 9 of the important paper by \textit{P. Bardsley} and \textit{R. W. Richardson} [Proc. Lond. Math. Soc. (3) 51, 295--317 (1985; Zbl 0604.14037)]. The paper is very well organized. After the introduction and the preliminaries occupying up to Section 3, Section 4 is devoted to the statement and the proof of the main Theorem 5, concerned with the map \((**)\). The remaining sections are devoted to apply the theorem in a number of concrete situations. First of all to the case where \(G=\mathrm{GL}_r\), so recovering the known results regarding the representations of the general linear group (surjection \((*)\)). Section 6 sets a few preliminaries necessary to study the case when \(G\) is chosen to be another classical group other than \(\mathrm{GL}_r\): the interplay with the theory of \(\lambda\)-rings is highlighted, here. Finally sections 7, 8, 9 are concerned with the specializations of the main Theorem 5, by fixing the group \(G\) to be respectively \(\mathrm{Sp}_{2n}\), \(\mathrm{SO}_{2n+1}\) and \(\mathrm{SO}_{2n}\). Section 10 studies the case in which \(G\) can be arbitrarily chosen among the classical groups, but assuming that the parabolic subgroup \(P\) coincides with the Borel subgroup \(B\). This difficult but exciting paper ends with a complete, though essential, reference list. Levi subgroups; polynomial representation; generalized flag varieties Kumar, S., Representation ring of Levi subgroups versus cohomology ring of flag varieties, \textit{Math. Ann.}, 366, 395-415, (2016) Homogeneous spaces and generalizations Representation ring of Levi subgroups versus cohomology ring of flag varieties

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(P_i(x)\) for \(1\leq i\leq n-1\) be the \(n\times n\)-matrix obtained from the \(n\times n\) identity matrix by placing the block \(\left( \begin{smallmatrix} x&1\\0&1\end{smallmatrix}\right)\) with \(x\) at the \((i,i)\)'th coordinate. Then the matrices \(P_i(x)\) satisfy the Coxeter relations \(P_i(x) P_j(y) =P_j(y) P_i(x)\) if \(|i-j|\geq 2\) and \(P_i(x) P_{i+1}(y) P_i(z) =P_{i+1}(z)P_i(y+xz) P_{i+1}(x)\). It is shown that, for any reduced decomposition \(i=(i_1, i_2, \dots , i_N)\) of a permutation \(w\) and any ring \(R\), there is a bijection \(P_i:(x_1,x_2, \dots , x_N) \to P_{i_1}(x_1) P_{i_2}(x_2) \cdots P_{i_N}(x_N)\) from \(\mathbb{R}^N\) to the Schubert cell of \(w\). Moreover, it is shown how to factor explicitly any element of the Schubert cell corresponding to \(w\) into a product of such matrices. Thus one obtains a parametrization of the Schubert cell. The formulas use planar configurations naturally associated to reduced decompositions. It is shown that the linear parts of these parametrizations give exactly all injective balanced labelings of the diagram of \(w\) [as defined by \textit{S. Fomin, C. Greene, V. Reiner} and \textit{M. Shimozono}, Eur. J. Comb. 18, No. 4, 373-389 (1997; Zbl 0871.05059)], and that the quadratic part characterizes the commutation classes of reduced decompositions. elementary matrices; Coxeter relations; Schubert cells; reduced decompositions; labelings of diagrams; planar configurations; factorization of matrices; commutation classes C. Kassel, A. Lascoux, and C. Reutenauer, ''Factorizations in Schubert cells,'' Adv. Math. 150 (2000), no. 1, 1--35. Factorization of matrices, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Factorizations in Schubert cells

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by work of \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)] on set-valued tableaux in relation to the \(K\)-theory of the Grassmannian, \textit{T. Lam} and \textit{P. Pylyavskyy} [``Combinatorial Hopf algebras and K-homology of Grassmanians'', Preprint, \url{arXiv:0705.2189}] studied six combinatorial Hopf algebras that can be thought of as \(K\)-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the \(K\)-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions. combinatorial Hopf algebra; \(K\)-theory; symmetric functions Patrias, R.: Antipode formulas for combinatorial Hopf algebras. Preprint arXiv:1501.00710 (2015) Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Antipode formulas for some combinatorial Hopf algebras

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study certain bijection between plane partitions and \(\mathbb{N}\)-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating functions are similar to classical MacMahon's formulas; one of these statistics is equidistributed with the usual volume. We also show natural connections with the longest increasing subsequences of words. plane partitions; MacMahon's formulas; dual Grothendieck polynomials; volume generating functions Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Combinatorial aspects of partitions of integers, Partitions of sets, Grassmannians, Schubert varieties, flag manifolds Enumeration of plane partitions by descents

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{P. Caldero} and \textit{A. Zelevinsky} [Mosc. Math. J. 6, No. 3, 411--429 (2006; Zbl 1133.16012)] studied the geometry of quiver Grassmannians for the Kronecker quiver and computed their Euler characteristics by examining natural stratification of quiver Grassmannians. We consider generalized Kronecker quivers and compute virtual Poincaré polynomials of certain varieties which are the images under projections from strata of quiver Grassmannians to ordinary Grassmannians. In contrast to the Kronecker quiver case, these polynomials do not necessarily have positive coefficients. The key ingredient is the explicit formula for noncommutative cluster variables given by \textit{R. Schiffler} and the first author [Compos. Math. 148, No. 6, 1821--1832 (2012; Zbl 1266.16027)]. quiver Grassmannians; ordinary Grassmannians; virtual Poincaré polynomials; Kronecker quiver; noncommutative cluster variables DOI: 10.1090/conm/592/11771 Cluster algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of quivers and partially ordered sets On natural maps from strata of quiver Grassmannians to ordinary Grassmannians

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This monograph is based on the author's lectures at the CBMS conference at North Carolina State University in 2001. It covers the application of symmetric functions to algebraic identities related to the Euclidean algorithm. It does not require extensive knowledge of symmetric functions, although some familiarity with the basic properties would be useful. Many texts on symmetric functions view the functions as polynomials. In constrast, this monograph uses the method of \(\lambda\)-rings, in which symmetric functions are viewed as operators on the ring of polynomials. This approach is particularly natural for these applications, because the construction of the symmetric functions and all necessary properties follow from a few fundamental results. An arbitrary polynomial can be viewed as a symmetric function in terms of its roots. In this framework, the successive remainders in the Euclidean algorithm can be expressed in terms of symmetric functions; examples include Sturm sequences and continued fractions. Divided difference operators also act on polynomials, and operations involving partial or full symmetrization can be expressed in terms of divided differences; one example is a symmetric function identity which arises in the cohomology of Grassmannians. Another viewpoint is that of orthogonal polynomials; if the ``moments'' are the complete symmetric functions, the resulting orthogonal polynomials are Schur functions indexed by square partitions, and other Schur functions appear in contexts such as Christoffel determinants. A similar approach can be used to study generalizations of symmetric functions. Schubert polynomials are constructed in two ways: by generalizing Newton's interpolation to multiple variables, and from a non-symmetric Cauchy kernel. The book concludes with a brief discussion of further generalizations to non-commutative Schur functions and Schubert polynomials. Schur functions; \(\lambda\)-rings; Cauchy kernel; Euclidean algorithm; continued fractions; Padé approximation; divided differences; cohomology of Grassmannian; orthogonal polynomials; Schubert polynomials Lascoux, A.: Symmetric functions \& combinatorial operators on polynomials. CBMS reg. Conf. ser. Math. 99 (2003) Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Approximation by polynomials, Padé approximation, Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis, Orthogonal polynomials [See also 33C45, 33C50, 33D45] Symmetric functions and combinatorial operators on polynomials

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We compute the Hilbert--Kunz functions and multiplicities for certain projective embeddings of flag varieties \(G/B\) and elliptic curves, over algebraically closed fields of positive characteristics. The group theoretic nature of both these classes of examples is used, albeit in different ways, to explicitly describe the cokernels in each degree of the Frobenius twisted multiplication maps for the corresponding graded rings. This detailed information also enables us to extend our results to arbitrary products of such varieties. \beginbarticle \bauthor\binitsN. \bsnmFakhruddin and \bauthor\binitsV. \bsnmTrivedi, \batitleHilbert-Kunz functions and multiplicities for full flag varieties and elliptic curves, \bjtitleJ. Pure Appl. Algebra \bvolume181 (\byear2003), page 23-\blpage52. \endbarticle \endbibitem Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Elliptic curves Hilbert--Kunz functions and multiplicities for full flag varieties and elliptic curves.

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors initiate a program to relate the geometry of affine Grassmannians with the representation theory of shifted Yangians. More precisely, they study slices in affine Grassmannians which correspond to weight spaces of irreducible representations under the geometric Satake correspondence. Their main result is that certain subquotients of Yangians quantize these slices. Let \(G\) be a complex semisimple group and consider its thick affine Grassmannian \(\mathrm{Gr}=G((t^{-1}))/G[t]\). Attached to each pair of dominant coweights \(\lambda \geq \mu\), we have Schubert varieties with \(\mathrm{Gr}^\mu\subset\overline{\mathrm{Gr}^\lambda}\subset \mathrm{Gr}\). The neighborhood in \(\overline{\mathrm{Gr}^\lambda}\) of a point in \(\mathrm{Gr}^\mu\) is encapsulated in a transversal slice, \(\mathrm{Gr}^{\overline{\lambda}}_\mu\), to the latter variety in the former. Under the geometric Satake correspondence, this slice is related to the \(\mu\) weight space in the irreducible representation of \(G^\vee\) of highest weight \(\lambda\). There is a natural structure of Poisson variety over \(\mathrm{Gr}\), such that the slice \(\mathrm{Gr}^{\overline{\lambda}}_\mu\) is an affine Poisson subvariety. The purpose of this paper is to describe quantization of the coordinate ring \(\mathcal{O}(\mathrm{Gr}^{\overline{\lambda}}_\mu)\) of \(\mathrm{Gr}^{\overline{\lambda}}_\mu\), which is a Poisson algebra. This algebra \(\mathcal{O}(\mathrm{Gr}^{\overline{\lambda}}_\mu)\) is a quotient of \(\mathcal{O}(\mathrm{Gr}_\mu)\), and \(\mathcal{O}(\mathrm{Gr}_\mu)\) is a subalgebra of \(\mathcal{O}(\mathrm{Gr}_1[[t^{-1}]]\). In order to quantize \(\mathrm{Gr}^{\overline{\lambda}}_\mu\), the authors follow a three-step procedure which mirrors this construction. First, they construct a version \(Y\) of the Yangian, which is a subalgebra of the Drinfeld Yangian. Next, they define subalgebras \(Y_\mu \subset Y\), called shifted Yangians, that quantize \(\mathrm{Gr}_\mu\). This generalizes the shifted Yangian for \(gl_n\) introduced by \textit{J. Brundan} and \textit{A. Kleshchev} [Adv. Math. 200, No. 1, 136--195 (2006; Zbl 1083.17006)]. Finally, they define a quotient \(Y^\lambda_\mu\) of \(Y_\mu\) using some remarkable representations of \(Y\) as difference operators, constructed in [\textit{A. Gerasimov} et al., Commun. Math. Phys. 260, No. 3, 511--525 (2005; Zbl 1142.53069)]. The authors conjecture that the scheme quantized by \(Y^\lambda_\mu\) is reduced. More precisely, they provide a conjectural description of the generators of the ideal of \(\mathrm{Gr}^{\overline{\lambda}}_\mu\) inside \(\mathrm{Gr}_\mu\) and prove that this conjecture implies that \(Y^\lambda_\mu\) quantizes the reduced scheme structure on \(\mathrm{Gr}^{\overline{\lambda}}_\mu\). Moreover, they prove that this conjecture gives a simple description for the ideal defining \(Y^\lambda_\mu\). This conjecture also implies the conjectural quantization of the Zastava spaces for \(\mathrm{PGL}_n\) of \textit{M. Finkelberg} and \textit{L. Rybnikov} [J. Eur. Math. Soc. (JEMS) 16, No. 2, 235--271 (2014; Zbl 1287.14024)]. quantization; affine Grassmannian; quantum groups; Yangian Kamnitzer, J.; Webster, B.; Weekes, A.; Yacobi, O., Yangians and quantizations of slices in the affine Grassmannian, Algebra Number Theory, 8, 857-893, (2014) Grassmannians, Schubert varieties, flag manifolds, Geometric Langlands program (algebro-geometric aspects), Deformation quantization, star products Yangians and quantizations of slices in the affine Grassmannian

In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a prime \(p\geq 2\), let \({A}_p\) be the mod-\(p\) Steenrod algebra and let \({P}^k \in {A}_p\) be the Steenrod mod-\(p\) reduced powers on the \(\mathbb Z_p\)-cohomology of topological spaces. Let \(G\) be a compact connected Lie group and let \(P\) be the centralizer of a one-parameter subgroup in \(G\). The space \(G/P\) of left cosets of \(P\) in \(G\) is a flag manifold. Let \(W\) and \(W'\) be the Weyl groups of \(G\) and \(P\), respectively. The set \(W / W'\) of the left cosets of \(W'\) in \(W\) can be identified with the following subset of \(W\): \[ \overline{W}= \{w \in W\mid l(w_1) \geq l(w)\, \text{for all}\,\, w_1 \in W'\} \] where \(l: W\rightarrow \mathbb Z\) is the length function. The flag space \(G/P\) has a canonical decomposition into cells \(X_w\) indexed by elements of \(\overline{W}\) where each cell \(X_w\) is the closure of Bruhat varieties, known as a Schubert variety in \(G/P\). Since cells are even dimensional, the set of fundamental classes \([X_w] \in H^{2l(w)}(G/P)\) forms an additive basis of the homology \(H_* (G/P)\). The cocycle \(\sigma_w \in H^{2l(w)} (G/P)\) defined by Kronecker pairing is called the Schubert class corresponding to \(w\). The set of Schubert classes \[ \{ \sigma_w \mid w \in \overline{W} \} \] forms an additive basis for the ring \(H^* (G/P)\). Then one has the following expression: \[ {P}^k (\sigma_u) \equiv \sum {a^k}_{u,w} \sigma_w\,\, \mod p, {a^k}_{u,w} \in \mathbb Z_p. \] In this work, the authors study the following problem on \(G/P\). Problem I: Determine the numbers \({a^k}_{u,w} \in \mathbb Z_p\) for \(k\geq 1, u,w \in \overline{W}\) with \(l (w) = l (u) + k(p-1)\). As a consequence of this work they present both a formula and an algorithm that determine the numbers \({a^k}_{u,w} \in \mathbb Z_p\) in terms of Cartan numbers of \(G\). Steenrod operations; Schubert calculus; flag manifolds; Schubert cells; Cartan numbers Duan, H B; Zhao, X Z, A unified formula for Steenrod operations in flag manifolds, Compos Math, 143, 257-270, (2007) Steenrod algebra, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Complete intersections A unified formula for Steenrod operations in flag manifolds