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Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes. (English) Zbl 1084.14049
This paper connects two degenerations related to the manifold $$F_n$$ of complete flags in $${\mathbb C}^n$$. N. Gonciulea and V. Lakshmibai [Transform. Groups 1, No. 3, 215–248 (1996; Zbl 0909.14028)] used standard monomial theory to construct a flat sagbi degeneration of $$F_n$$ into the toric variety of the Gelfand-Tsetlin polytope, and recently A. Knutson and E. Miller [Ann. Math. (2) 161, No. 3, 215–248 (2005; Zbl 1089.14007)] constructed Gröbner degenerations of matrix Schubert varieties into linear spaces corresponding to monomials in double Schubert polynmials. The flag variety is the geometric invariant theory (GIT) quotient of the space $$M_n$$ of $$n$$ by $$n$$ matrices by the Borel group $$B$$ of lower triangular matrices. A matrix Schubert variety is an inverse image of a Schubert variety under this quotient. The main result in the paper under review is that this GIT quotient extends to the degenerations. The sagbi degeneration is a GIT quotient of the Gröbner degeneration.\smallskip The nature of this GIT quotient is quite interesting. The authors exhibit an action of the Borel group $$B$$ on the product $$M_n\times{\mathbb C}$$ of $$M_n$$ with the complex line, so that the GIT quotient $$B\backslash\backslash(M_n\times{\mathbb C})$$ remains fibred over $${\mathbb C}$$ and is the total space of the sagbi degeneration. In this GIT quotient, the total space of the Gröbner degeneration (as in Knutson and Miller) of a matrix Schubert variety in $$M_n\times{\mathbb C}$$ covers the total space of the Lakshmibai-Gonciulea degeneration of the corresponding Schubert variety. At the degenerate point, the matrix Schubert variety has become a union of coordinate planes, each of which covers a component of the sagbi degeneration of the Schubert variety indexed by a face of the Gelfand-Tsetlin polytope.
The authors use this to identify which faces of the Gelfand-Tsetlin polytope occur in a given degenerate Schubert variety, and to give a simple explanation of the classical Gelfand-Tsetlin decomposition of an irreducible polynomial representation of $$\text{GL}_n$$ into one-dimensional weight spaces; in the degeneration, sections of a line bundle over $$F_n$$ become sections of the defining line bundle on the toric variety for the Gelfand-Tsetlin polytope.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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##### References:
 [1] Bergeron, N.; Billey, S., RC-graphs and Schubert polynomials, Exp. math., 2, 4, 257-269, (1993) · Zbl 0803.05054 [2] Buch, A.S.; Fulton, W., Chern class formulas for quiver varieties, Invent. math., 135, 3, 665-687, (1999) · Zbl 0942.14027 [3] Billey, S.C.; Jockusch, W.; Stanley, R.P., Some combinatorial properties of Schubert polynomials, J. algebraic combin., 2, 4, 345-374, (1993) · Zbl 0790.05093 [4] Buch, A.S., Grothendieck classes of quiver varieties, Duke math. J., 115, 1, 75-103, (2002) · Zbl 1052.14056 [5] Caldero, P., Toric degenerations of Schubert varieties, Transform. groups, 7, 1, 51-60, (2002) · Zbl 1050.14040 [6] Chirivı̀, R., LS algebras and application to Schubert varieties, Transform. groups, 5, 3, 245-264, (2000) · Zbl 1019.14019 [7] Eisenbud, D., Commutative algebra, with a view toward algebraic geometry, Graduate texts in mathematics, Vol. 150, (1995), Springer New York · Zbl 0819.13001 [8] Fomin, S.; Kirillov, A.N., Combinatorial Bn-analogues of Schubert polynomials, Trans. amer. math. soc., 348, 9, 3591-3620, (1996) · Zbl 0871.05060 [9] Fomin, S.; Kirillov, A.N., The Yang-Baxter equation, symmetric functions and Schubert polynomials, Discrete math., 153, 1-3, 123-143, (1996), Proceedings of the Fifth Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993 · Zbl 0852.05078 [10] Fomin, S.; Stanley, R.P., Schubert polynomials and the nil-Coxeter algebra, Adv. math., 103, 2, 196-207, (1994) · Zbl 0809.05091 [11] Fulton, W., Flags, Schubert polynomials, degeneracy loci and determinantal formulas, Duke math. J., 65, 3, 381-420, (1992) · Zbl 0788.14044 [12] Fulton, W., Introduction to toric varieties, Annals of mathematical studies, Vol. 131, (1993), Princeton University Press [13] Fulton, W., Young tableaux. with applications to representation theory and geometry, London mathematical society student texts, Vol. 35, (1997), Cambridge University Press Cambridge · Zbl 0878.14034 [14] Fulton, W., Universal Schubert polynomials, Duke math. J., 96, 3, 575-594, (1999) · Zbl 0981.14022 [15] Gonciulea, N.; Lakshmibai, V., Degenerations of flag and Schubert varieties to toric varieties, Transform. groups, 1, 3, 215-248, (1996) · Zbl 0909.14028 [16] Guillemin, V.; Sternberg, S., The Gel’fand-cetlin system and quantization of the complex flag manifolds, J. funct. anal., 52, 1, 106-128, (1983) · Zbl 0522.58021 [17] Gelfand, I.M.; Tsetlin, M.L., Finite-dimensional representations of the group of unimodular matrices, Dokl. akad. nauk SSSR (N.S.), 71, 825-828, (1950) · Zbl 0037.15301 [18] A. Knutson, E. Miller, Gröbner geometry of Schubert polynomials, Ann. Math. (2) (2004) to appear. arXiv:math.AG/0110058v3. [19] A. Knutson, E. Miller, M. Shimozono, Four positive formulae for quiver polynomials, 2003 preprint. arXiv:math.AG/0308142. · Zbl 1107.14046 [20] M. Kogan, Schubert geometry of flag varieties and Gel’fand-Cetlin theory, Ph.D. Thesis, Massachusetts Institute of Technology, 2000. [21] Lascoux, A.; Schützenberger, M.-P., Polynômes de Schubert, C. R. acad. sci. Paris Sér. I math., 294, 13, 447-450, (1982) · Zbl 0495.14031 [22] Lascoux, A.; Schützenberger, M.-P., Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux, C. R. acad. sci. Paris Sér. I math., 295, 11, 629-633, (1982) · Zbl 0542.14030 [23] Littelmann, P., Cones, crystals, and patterns, Transform. groups, 3, 2, 145-179, (1998) · Zbl 0908.17010 [24] Sturmfels, B., Gröbner bases and convex polytopes, AMS university lecture series, Vol. 8, (1996), American Mathematical Society Providence, RI · Zbl 0856.13020 [25] R. Vakil, A geometric Littlewood-Richardson rule. arXiv:math.AG/0302294. · Zbl 1163.05337 [26] R. Vakil, Schubert induction. arXiv:math.AG/0302296. · Zbl 1115.14043
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