TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS:
THE CASE OF SQUARE MATRICES

arXiv:1608.04058v2 [] 8 Jan 2018

NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM
Abstract. We characterize the cone of GL-equivariant Betti tables of Cohen–Macaulay
modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for ‘Boij–Söderberg theory for Grassmannians’,
with the goal of characterizing the cones of GLk -equivariant Betti tables of modules over
the coordinate ring of k ˆ n matrices, and, dually, cohomology tables of vector bundles
on the Grassmannian Grpk, Cn q. The proof uses Hall’s Theorem on perfect matchings in
bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding
equivariant free resolutions by applying Weyman’s geometric technique to certain graded
pure complexes of Eisenbud–Fløystad–Weyman.

1. Introduction
1.1. Ordinary and equivariant Boij–Söderberg theory. Let M be a finitely-generated
Z-graded module over a polynomial ring A “ Crx1 , . . . , xn s. The Betti table of M counts
the number of generators in each degree of a minimal free resolution of M. More precisely,
if M has a graded minimal free resolution of the form
à
à
Ap´dqβnd Ð 0,
Ap´dq‘β0d Ð ¨ ¨ ¨ Ð
MÐ
dPZ

dPZ

the Betti table of M is the collection of numbers βij . Equivalently, βij is the dimension of
the degree-j part of the graded module TorA
i pM, Cq.
Boij–Söderberg theory (initiated in [BS08]) seeks to characterize the possible Betti tables
of graded modules over polynomial rings, with the key insight that it is easier to study these
tables only up to positive scalar multiple. The theory has been broadly successful: while
the earliest results concerned Betti tables of Cohen–Macaulay modules (stratified by their
codimension) [EFW11, ES09], the theory was extended to all modules [BS12], to certain
modules over multigraded and toric rings [BBEG12, EE12], and more [KS15, GS16]. For
some surveys, see [Flø12, FMP16].
In fact, the classification is surprisingly simple. We say a Betti table is pure if, for each
i, exactly one βij is nonzero, i.e., each step of the minimal free resolution is concentrated
in a single degree. For each increasing sequence of integers d0 ă d1 ă ¨ ¨ ¨ ă dk , there is
a Cohen–Macaulay A-module whose Betti table is pure with the βidi ’s as the only nonzero
entries. Moreover, the resulting Betti table is unique up to a rational multiple, and any
purported Betti table is a positive rational multiple of the Betti table of an actual module
if and only if it can be written as a positive Q-linear combination of pure tables [ES09]. If
we bound the degrees that can occur in a Betti table — that is, bound the j’s for which βij
Date: January 7, 2018.
2010 Mathematics Subject Classification. Primary 13D02, Secondary 05E99.
Key words and phrases. Boij–Söderberg theory, Betti table, cohomology table, Schur functor, equivariant
K-theory.
1

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NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

may be nonzero — the Betti tables that fit within these bounds form a rational polyhedral
cone. The pure tables then form the extremal rays of this cone.
A key feature of the theory is the discovery that the cone of Betti tables is dual to another
cone, consisting of cohomology tables of vector bundles and sheaves on projective space.
Given such a sheaf F, its cohomology table is the table of numbers γij “ hi pF b Opjqq. There
is a family of nonnegative bilinear pairings between Betti tables of modules and cohomology
tables of sheaves, and the inequalities that cut out the cone of Betti tables can all be realized
explicitly in terms of this pairing. Consequently, Betti tables yield numerical constraints on
the possible cohomology tables on Pn , and vice versa. Recent work of Eisenbud–Erman
has categorified this pairing [EE12], realizing it through a functorial pairing between the
underlying algebraic objects.
This paper is the beginning of an attempt to generalize this story to GLk -equivariant
modules over a polynomial ring (all GLk -modules are required to be algebraic representations). Write R “ Crx11 , x12 , . . . , xkn s, the coordinate ring of the affine space of k ˆ n
matrices with the left GLk action. In this setting, as we will see in Section 2, a minimal
free resolution of a finitely-generated equivariant R-module comes with an action of GLk ,
so in forming our Betti tables we can ask which representations appear at each step of the
resolution rather than just which degrees. Specifically, by analogy with the ordinary case,
we wish to understand:
(i) the cone BSk,n of GLk -equivariant Betti tables of modules supported on the locus
of rank-deficient matrices in HompCk , Cn q, and
(ii) the cone ESk,n of GL-cohomology tables of vector bundles on the Grassmannian
Grpk, Cn q.
Both of these constructions will be defined more precisely in Section 2.
Remark 1.1. The case k “ 1 is the ordinary Boij–Söderberg theory of graded modules and
vector bundles on projective space since an algebraic action of GL1 is equivalent to a choice of
Z-grading (see Remark 2.1). We should point out that in this case, [SW11] studies a GLn equivariant analogue of Boij–Söderberg theory using a Schur-positive analogue of convex
cones. In [SW11], the equivariant Betti table records characters and the Boij–Söderberg
cone is defined to be closed under “Schur positive rational functions” while in the current
work, the equivariant Betti table records multiplicities and the cone has an action of the
positive rational numbers instead.

In a later paper, we establish a nonnegative pairing between these tables, extending the
pairing of Eisenbud–Schreyer; the hope is that the cones are dual, as they are in ordinary
Boij–Söderberg theory. On the algebraic side, we will restrict to Cohen–Macaulay modules
supported everywhere along the rank-deficient locus.
A fundamental base case in ordinary Boij–Söderberg theory is to understand Betti tables
of torsion graded modules over a polynomial ring Crts in one variable. The categorified
pairing of Eisenbud–Erman effectively outputs such a module (actually a complex of such
modules); as such, the structure of these tables, while very simple, controls the structure
of the general Boij–Söderberg cone and its dual. It is relatively straightforward to write
down the inequalities that cut out the corresponding cone, and every inequality cutting out
the larger cone of Betti tables comes from pulling back one of these through the pairing
mentioned earlier.

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

3

This paper is concerned with the corresponding base case, namely, the cone of equivariant
modules over the coordinate ring of square matrices. This case looks simple at first glance:
the modules have codimension 1, and the corresponding Grassmannian is just a point, so
there is no dual picture involving vector bundles. Unlike in the graded setting, however,
the equivariant base case is already both combinatorially and algebraically interesting. Our
main result is the following description of this cone:
Theorem 1.2. In the square matrix case, the supporting hyperplanes of the equivariant
Boij–Söderberg cone BSk,k correspond to antichains in the extended Young’s lattice Y˘ of
weakly-decreasing integer sequences. Its extremal rays correspond to pure resolutions and
are indexed by comparable pairs of weakly-decreasing integer sequences, λp0q Ĺ λp1q .
For a more precise version of this statement, see Theorem 3.8.
We will exhibit a free resolution to realize each extremal ray, but the construction is
nontrivial and relies on existing results of Eisenbud–Fløystad–Weyman from ordinary Boij–
Söderberg theory. The proof presented here also depends crucially on the Borel–Weil–Bott
theorem, so we do not know if our results hold in positive characteristic.
We expect the description of the cone in Theorem 1.2 to control the structure of the
equivariant Boij–Söderberg cone in the general case. In particular, the generalized Eisenbud–
Schreyer pairing will map the larger cones BSk,n and ESk,n to the square-matrix cone. We
sketch this construction in Section 2.
1.2. Structure of the paper. The paper is structured as follows: in Section 2, we introduce
the relevant notions, namely, equivariant Betti tables and (briefly) GL-cohomology tables
for sheaves on Grassmannians. In Section 3, we describe the combinatorics of the equivariant
Boij–Söderberg cone for square matrices. In Section 4, we show that each extremal ray is
realizable, using Weyman’s geometric technique.
1.3. Acknowledgments. We thank Daniel Erman, Maria Gillespie, and David Speyer for
helpful conversations. Computations in SAGE [Dev16] and Macaulay2 [GS] have also been
very helpful for this work. Steven Sam was partially supported by NSF DMS-1500069.
2. Setup
Throughout, let V, W be vector spaces over C of dimensions k, n respectively, with k ď n.
Starting in Section 3, we will assume n “ k.
2.1. Background. We will only consider algebraic representations of GLpV q. A good introduction to these notions is [Ful96]. We will also refer the reader to [SS12, §3] for a succinct
summary (with references) of what we’ll need about the representation theory of the general
linear group.
The irreducible (algebraic) representations of GLpV q are indexed by weakly-decreasing
integer sequences λ “ pλ1 , . . . , λk q, where k “ dimpV q. We write Sλ pV q for the corresponding
representation, called a Schur functor. If λ has all nonnegative parts, we write λ ě 0 and
say λ is a partition. We often represent partitions by their Young diagrams:
λ “ p3, 1q ÐÑ λ “

.

We partially order partitions and integer sequences by containment:
λ Ď µ if λi ď µi for all i.

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NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

We write Y for the poset of all partitions with this ordering, called Young’s lattice. We write
Y˘ for the set of all weakly-decreasing integer sequences; we call it the extended Young’s
lattice.
If λ is a partition, Sλ pV q is functorial for linear transformations V Ñ W . If λ has
„
negative parts, Sλ is only functorial for isomorphisms V Ý
Ñ W . If λ “ pd, 0, . . . , 0q, then
Ź
Sλ pV q “ Symd pV q and if λ “ 1d “ p1, . . . , 1, 0, . . . 0q with d 1’s, then Sλ pV q “ d pV q. If
Ź
dim V “ k, we’ll write detpV q for the one-dimensional representation k pV q “ S1k pV q. We
write Kλ pkq for the dimension of Sλ pCk q.
We may always twist a representation by powers of the determinant:
detpV qba b Sλ1 ,...,λk pV q “ Sλ1 `a,...,λk `a pV q
for any integer a P Z. This operation is invertible and can sometimes be used to reduce to
considering the case when λ is a partition.
By semisimplicity, any tensor product of Schur functors is isomorphic to a direct sum of
Schur functors with some multiplicities:
à
ν
Sλ pV q b Sµ pV q –
Sν pV q‘cλ,µ .
ν

cνλ,µ

The
are the Littlewood–Richardson coefficients; we won’t need to know how they are
computed in general, though we will use that if cνλ,µ ‰ 0 and λ is a partition, then µ Ď ν
(and similarly, if µ is a partition, then λ Ď ν). Also, by symmetry of tensor products, we
have cνλ,µ “ cνµ,λ . An important special case is Pieri’s rule when λ “ pdq. In this case,
cνpdq,µ ď 1 and is nonzero if and only if µ Ď ν and the complement of µ in ν is a horizontal
strip, i.e., does not have more than 1 box in any column. This is equivalent to the interlacing
inequalities ν1 ě µ1 ě ν2 ě µ2 ě ¨ ¨ ¨ .
If R is a C-algebra with an action of GLpV q, and S is any GLpV q-representation, then
S bC R is an equivariant free R-module; it has the universal property
HomGLpV q,R pS bC R, Mq – HomGLpV q pS, Mq
for all equivariant R-modules M. The basic examples will be the modules Sλ pV q b R.
Remark 2.1 (Gradings and GL1 ). If dimpV q “ 1, the notion of GLpV q-equivariant ring or
module is identical to ‘(Z-)graded’. In particular, in this case V bd b R – Rp´dq, the rank-1
free module generated in degree d.
In general, the modules Sλ pV q b R are the equivariant analogues of the twisted graded
modules Rp´dq. The analogous notion to ‘N-graded ring’ is that, as a GLpV q-representation,
R should contain only those Sλ with nonnegative parts.

2.2. Equivariant modules and Betti tables. Fix two vector spaces V and W with k “
dimpV q ď dimpW q “ n. Let X be the affine variety HompV, W q, with coordinate ring

„
1 ď i ď k,
˚
˚
.
R “ OX “ SympHompV, W q q “ SympV b W q – C xij :
1ďjďn
The ring R has actions of GLpV q and GLpW q. Its structure as a representation is given by
the Cauchy identity (see [SS12, (3.13)] for example):
à
(2.1)
R–
Sν pV q b Sν pW ˚ q.
νě0

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

5

We are primarily interested in the GLpV q action, though we will use both actions when we
construct resolutions in Section 4.
The rank-deficient locus tT : kerpT q ‰ 0u Ă X is` an
˘ irreducible subvariety of codimension
n ´ k ` 1. Its prime ideal Pk is generated by the nk maximal minors of the k ˆ n matrix
pxij q. When k “ n, Pk is a principal ideal, generated by the determinant.
Note that the maximal ideal m “ pxij q of the origin in X and the ideal Pk are GLpV qand GLpW q-equivariant.
Let M be a finitely-generated GLpV q-equivariant R-module. The module ToriR pR{m, Mq
naturally has the structure of a finite-dimensional GLpV q-representation. We define the
equivariant Betti number βi,λ pMq as the multiplicity of the Schur functor Sλ pV q in this
Tor module, i.e.
à
ToriR pR{m, Mq –
Sλ pV q‘βi,λ pM q
(as GLpV q-representations).
λ

It is convenient also to define the (equivariant) rank Betti number Ą
βi,λ pMq as the dimension
of the λ-isotypic component, that is,
Ą
βi,λ :“ βi,λ ¨ dimC pSλ pV qq “ βi,λ ¨ Kλ pkq.

By semisimplicity of GLpV q-representations, it is easy to see that any minimal free resolution
of M can be made equivariant, so we may instead define βi,λ as the multiplicity of the
equivariant free module Sλ pV q b R in the i-th step of an equivariant minimal free resolution
of M:
à
M Ð F0 Ð F1 Ð ¨ ¨ ¨ Ð Fd Ð 0, where Fi “
Sλ pV qβi,λ pM q b R,
λ

and likewise Ą
βi,λ is the rank of the corresponding summand as an R-module.

Remark 2.2. Both definitions βi,λ , Ą
βi,λ are useful. The Betti number is needed for the
pairing with vector bundles, but the rank Betti number is more relevant to the square
matrix case and will play the more significant role in this paper.

2.2.1. Betti tables and cones. Let
Bk,n “

n´k`1
à à
i“0

Qi,λ

λ

be a direct sum of copies of Q, indexed by homological degree i and partition λ. We think
of an element of Bk,n as an abstract Betti table, that is, a choice of βi,λ for each i and λ.
r k,n for the space of abstract rank Betti tables pĄ
Similarly, we write B
βi,λ q.
Ă k,n Ď B
r k,n as the
We define the equivariant Boij–Söderberg cones BSk,n Ď Bk,n , BS
positive linear span of all (multiplicity or rank) Betti tables of finitely generated Cohen–
Macaulay a
modules M supported on the rank-deficient locus Spec R{Pk Ă X. (That is, M
for which annpMq “ Pk .)

2.3. GL-cohomology tables for Grassmannians. Let Grpk, W q denote the Grassmannian variety of k-dimensional subspaces in W , with tautological exact sequence of vector
bundles
0ÑSÑW ÑQÑ0

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NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

where W denotes the trivial rank-n vector bundle and
S “ tpx, Uq P W ˆ Grpk, W q : x P Uu.
Let E be any coherent sheaf on Grpk, W q. We define the GL-cohomology table γi,λpEq by
We let ESk,n Ă

À ś
i

γi,λ pEq :“ dimC Hi pE b Sλ pSqq.
λ

Qi,λ be the positive span of such tables.

Remark 2.3. Note that if k “ 1 then S “ Op´1q on the projective space PpW q. Since this
is a line bundle, λ can have only one row, say λ “ pjq. Then
Sλ pSq “ Symj pOp´1qq “ Op´jq,
so γi,λ pEq “ γi,´j pEq is the usual cohomology table of E with respect to Op1q. In general,
the GL-cohomology table contains more information than the usual cohomology table with
respect to twists by Op1q; in particular, it determines the class of E in K-theory KpGrpk, W qq,
Ź
while the usual table only determines the K-class of i˚ pEq, where i : Grpk, W q ãÑ Pp k pW qq
is the Plücker embedding.

2.4. The numerical pairing. We briefly discuss the pairing between equivariant Betti
tables and GL-cohomology tables. For details, see [FL16]. Let B “ pβi,λ q be an equivariant
r
Betti table and Γ “ pγi,λq a GL-cohomology table. We define a rank table ΦpB,
Γq “
`
˘
Ą
φi,λpB, Γq , for i P Z, by
ÿ
Ą
φ
βp,λ ¨ γq,λ.
i,λ pB, Γq “
p´q“i

In this definition we do not assumeÀany À
bounds on i, so it is convenient to define the
D
derived Boij–Söderberg cone BSk,n Ă iPZ λ Qi,λ as the positive linear span of Betti tables
À
β‚,λ
of minimal free equivariant complexes F ‚ “
b R with homology modules
λ Sλ pV q
supported in the rank-deficient locus.
r defines a pairing of cones,
Theorem 2.4 ([FL16, Theorem 1.13]). The map Φ
r : BSD ˆ ESk,n Ñ BS
ĂD .
Φ
k,n
k,k

ĂD
Consequently, any nonnegative linear functional on the cone BS
k,k determines a nonnegative bilinear pairing between equivariant Betti tables and GL-cohomology tables. The
ĂD
Ă
extended cone BS
k,k has extremal rays and facets closely resembling those of BSk,k . See
[FL16, Section 4.2] for an explicit description.
3. The Boij–Söderberg cone on square matrices
We now assume V, W are vector spaces of the same dimension k, and we describe the
r k,k . In particular, we would like to know both the extremal rays and the
Ă k,k Ă B
cone BS
equations of the supporting hyperplanes. The modules M of interest are Cohen–Macaulay
of codimension 1, so their minimal free resolutions are just injective maps F1 ãÑ F0 of
equivariant free modules. For i “ 0, 1, we put
à
Sλ pV qβi,λ b R,
Fi “
λ

Ą
and define β
i,λ “ βi,λ ¨ Kλ pkq as in Section 2.

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

7

The first observation is that, since M is a torsion module, we must have
ÿ
ÿ
βĄ
βĄ
(3.1)
rank F0 “ rank F1 , that is,
1,λ pMq.
0,λ pMq “
λ

λ

Conversely, any injective map of free modules of this form must have a torsion cokernel,
which is then Cohen–Macaulay of codimension 1. We will see that the rank condition is the
only linear constraint on Betti tables, that is, the cone spans this entire linear subspace.
3.1. Antichain inequalities. The maps of any minimal complex have positive degree.
More precisely, we have the following:

Lemma 3.1 (Sequences contract under minimal maps). Let f : Sµ pV q b R Ñ Sλ pV q b R be
any nonzero map. If µ “ λ, then f is an isomorphism. Otherwise, µ Ľ λ and f is minimal.
Proof. This follows from the universal property of equivariant free modules,
HomGLpV q,R pSµ pV q b R, Sλ pV q b Rq – HomGLpV q pSµ pV q, Sλ pV q b Rq.
We apply the Cauchy identity (2.1) for R as a GLpV q-representation. We see that
à
`
˘
HomGLpV q pSµ pV q, Sλ pV q b Rq –
HomGLpV q Sµ pV q, Sλ pV q b Sν pV q b Sν pW ˚ q.
νě0

By the Littlewood–Richardson rule, if µ Ğ λ, every summand is 0. If µ “ λ, the only
nonzero summand comes from ν “ ∅; we see that the corresponding map is an isomorphism
(if nonzero). Finally, if µ Ľ λ, there is at least one ν for which the corresponding summand
is nonzero and any such ν must satisfy |ν| “ |µ| ´ |λ| ą 0, so the corresponding map of
R-modules has strictly positive degree (equal to |µ|), hence is a minimal map.

Remark 3.2. Because the ring R involves W ˚ , not W , the analogous computation shows
that the sequence labeling W expands under a minimal map: that is, a nonzero GLpW qequivariant map Sµ pW q b R Ñ Sλ pW q b R exists if and only if µ Ď λ (and is minimal if and
only if µ ‰ λ).

In particular, for any fixed µ, a minimal injective map F1 ãÑ F0 of free modules must inject
the summands λ Ď µ of F1 into the summands λ Ĺ µ of F0 , and so
ÿ
ÿ
βĄ
βĄ
ě
(3.2)
1,λ ,
0,λ
λĹµ

λĎµ

which gives us some of the inequalities our Betti tables need to satisfy. But in fact these
inequalities are not enough. For example, for any pair of partitions α, β, the summands of
F1 given by
tλ : λ Ď α or λ Ď βu
must inject into the summands of F0 given by
tλ : λ Ĺ α or λ Ĺ βu.
This gives the additional, non-redundant condition
ÿ
ÿ
Ą
β
0,λ ě
λĹα or λĹβ

λĎα or λĹβ

βĄ
1,λ .

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NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

Example 3.3. The following example illustrates that the inequalities (3.2) are not sufficient.
Consider the following rank Betti table, with all entries equal to 1 (shown transposed, with
dashed lines indicating containment of partitions):
βĂ
0λ :
βĂ
1λ :

✞

✞

✞

✞

✞

❁

❁

❁

❁

❁

❁

It is evident that this cannot be the Betti table of a torsion module, since nothing maps to
the

summand. Likewise, the table violates the inequality for the pair

,

. For

any single partition µ, however, the inequality (3.2) is satisfied. (Note that if µ contains
both

and

, then it strictly contains

.)



The complete set of inequalities is as follows. Recall that if P is a poset, I Ď P is an order
ideal if x P I and y ď x implies that y P I, i.e., I is a downwards-closed subset. We define
the interior of I to be the subset
I ˝ “ tx P P : x ă y for some y P Iu
of elements strictly contained in I. The maximal elements IzI ˝ of I form an antichain, that
is, they are pairwise incomparable. We have the following:
Lemma 3.4 (Antichain inequalities). Let Y˘ be the extended Young’s lattice and I Ď Y˘
an order ideal. Let pβi,λ q P BSk,k be a Betti table. Then
ÿ
ÿ
βĄ
βĄ
(3.3)
1,λ .
0,λ ě
λPI ˝

λPI

Proof. Follows from the above discussion.



Remark 3.5 (Inequalities for upwards-closed sets). It is also the case that, for any upwardsclosed subset U Ď Y˘ , we have a “dual” inequality
ÿ
ÿ
βĄ
βĄ
ď
(3.4)
1,λ ,
0,λ
λPU˝

λPU

where U˝ “ tλ P U : λ ě µ for some µ P Uu is its upwards-interior.
Algebraically, this corresponds to the following observation: let pF1 qU˝ , pF0 qU be the summands corresponding to U˝ , U. The projection F0 ։ pF0 qU vanishes on the images of the
non-U˝ summands of F1 , so we have a commutative diagram
F1


pF1 qU˝

f

f¯

/

F0


/

pF0 qU .

It follows that cokerpf q Ñ cokerpf¯q is surjective, hence cokerpf¯q is also torsion (since cokerpf q
is). Consequently, we obtain the desired inequality rankpF1 qU˝ ě rankpF0 qU .

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

9

Alternatively, we may deduce (3.4) by subtracting the inequality (3.3) with I “ Y˘ zpU ˝ q
from the rank equation (3.1), and observing that
pP zpU˝ qq˝ Ď P zU
holds in any poset P . (Note that the complement of an upwards-closed set is downwardsclosed.) In particular, given the rank equation (3.1), the “upwards-facing” and “downwardsfacing” inequalities collectively cut out the same cone.

3.2. Extremal rays and pure diagrams. We can construct a very simple Betti table by
Č
Č
letting λp0q Ĺ λp1q be any pair of distinct, comparable partitions. Let β
0,λp0q “ β1,λp1q “ 1
and let all other entries of the Betti table be 0. We call the resulting table Prpλp0q , λp1q q a
pure table of type pλp0q , λp1q q, and any resolution corresponding to a positive multiple of
this table a pure resolution.
Note that a pure table clearly satisfies all of the antichain inequalities (3.3), as well as the
linear constraint (3.1). Moreover, since a Betti table must have at least two nonzero entries,
a pure table cannot be written as a nontrivial positive combination of other tables. Any
Ă k,k .
realizable pure table therefore generates an extremal ray of BS
We will show in Theorem 4.1 that every pure table has a realizable scalar multiple. ConĂ k,k .
sequently, every pure table generates an extremal ray of the Boij–Söderberg cone BS
We now show that, assuming Theorem 4.1, every extremal ray is of this form.

Theorem 3.6 (Extremal rays). Every realizable rank Betti table is a positive Z-linear combination of pure rank tables.
The proof uses Hall’s Theorem on perfect matchings in bipartite graphs. Recall that a
perfect matching on a graph G is a subset E 1 Ď E of the edges of G, such that every
vertex of G occurs on exactly one edge from E 1 . We recall the statement of Hall’s Matching
Theorem (see [LP09, Theorem 1.1.3] for a proof):
Theorem 3.7 (Hall). Let G be a bipartite graph with left vertices L, right vertices R and
edges E. For a collection of vertices S, let ΓpSq be the set of neighboring vertices to S.
Assume |L| “ |R|. Then G has a perfect matching if and only if |ΓpSq| ě |S| for all
subsets S Ď R.
Ă k,k be a realizable rank Betti table; by rescaling, we
Proof of Theorem 3.6. Let pĄ
βi,λ q P BS
may assume all the entries are integers. We define a bipartite graph G “ pL, R, Eq as follows:
Ą
for each λ, L (resp. R) will have βĄ
0,λ vertices (resp. β1,λ ) labeled λ. Every vertex labeled
λ in L is connected to every vertex labeled µ in R whenever λ Ĺ µ. By the rank condition
(3.1), G satisfies |L| “ |R|.
Observe that a perfect matching on G decomposes pβi,λ q as a Z-linear combination of pure
tables: each edge pλp0q Ð λp1q q in the matching corresponds to a pure table Prpλp0q , λp1q q.
Thus, it suffices to show that G has a perfect matching.
We apply Hall’s Theorem (Theorem 3.7). Let S Ď R. Observe that if S contains a vertex
labeled µ, then without loss of generality, we may assume S contains every vertex labeled µ
and, in addition, every vertex labeled µ1 with µ1 Ď µ, since adding these vertices makes S
larger but does not change ΓpSq.
Let I be the order ideal generated by the set of vertex labels appearing in S. We see
ř Ą
ř
Ą
that |S| “ λPI β
1,λ and |ΓpSq| “
λPI ˝ β0,λ , so the condition |ΓpSq| ě |S| is precisely the
antichain inequality (3.3) for I.


10

NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

Thus, assuming Theorem 4.1, we have shown:
Ă k,k ). The cone BS
Ă k,k Ă B
r k,k is cut out by the
Theorem 3.8 (Combinatorial description of BS
rank equation (3.1), the antichain inequalities (3.3) and the conditions βri,λ ě 0. Its extremal
rays are exactly the rays spanned by the pure tables corresponding to all pairs λp0q Ĺ λp1q of
comparable elements of Y˘ .
Remark 3.9 (Decomposing Betti tables). There are efficient algorithms for computing perfect matchings of graphs; see e.g. [LP09, §1.2]. A standard proof of Hall’s Theorem implicitly
r P BS
Ă k,k be a
uses the following algorithm, which is inefficient but conceptually clear. Let B
Betti table.
Case 1: Suppose every antichain inequality is strict. Choose any pure table Prpλp0q , λp1q q whose
r Then
entries occur with nonzero values in B.
rrest “ B
r ´ Prpλp0q , λp1q q P BS
Ă k,k .
B

rrest .
Continue the algorithm on B
Case 2: Suppose, instead, there exists an antichain I for which (3.3) is an equality. Write
r“B
rI ` B
rrest ,
B

˝
Ą
rI contains all the entries involved in the equality (βĄ
where B
0,λ for λ P I and β1,λ for
rI P BS
Ă k,k and B
rrest P BS
Ă k,k ; continue the algorithm separately
λ P I). Then both B
for each.

We contrast the algorithm above with the usual algorithm [ES09, §1] for decomposing graded
Betti tables. For graded tables, the decomposition is “greedy” and deterministic. It relies on
a partial ordering on pure graded Betti tables, which induces a decomposition of the Boij–
Söderberg cone as a simplicial fan. Unfortunately, the natural choices of partial ordering on
the equivariant pure tables P pλp0q , λp1q q do not yield valid greedy decomposition algorithms.
For example, suppose the graph G of Theorem 3.6 consists of a single long path. Compare
the following two examples:
βĂ
0λ :

❂

❂

❂

❂

⑧

⑧

⑧

⑧

⑧

❄

❄

❄
❄

❄

❄

❄

❄

❄

❄

⑤

⑤

⑤

⑤

⑤

❇

❇

❇

❇

βĂ
1λ :

In both cases, G has a unique perfect matching, but whether an edge is used depends on
its placement along the path, not just on the partitions labelling its vertices. Hence, an
algorithm that (for instance) greedily selects the lexicographically-largest pair pλp0q , λp1q q
will fail: in both cases, the lex-largest λp0q is
“ p3, 1q and its lex-largest neighbor is
“ p3, 2q. This leads to the (unique) correct matching on the first graph, but
λp1q “
fails on the graph to the right.
Similarly, the graph structure of G is a cycle, then G has two perfect matchings, so a
deterministic algorithm must have a way of selecting one.

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

11

Finally, unlike in the graded case, we do not know a good simplicial decomposition of
Ă
BSk,k ; it would be interesting to find such a structure.

4. Constructing Pure Resolutions
The main theorem of this section is as follows.
Theorem 4.1. For any partitions λp0q Ĺ λp1q , there exists a torsion, GLpV q-equivariant
R-module M with minimal free resolution
M Ð Sλp0q pV qc0 b R Ð Sλp1q pV qc1 b R Ð 0,
for some positive integers c0 , c1 .
We first consider a pair of partitions differing by a box. The same argument works somewhat more generally (see Remark 4.8), but we restrict to this case for notational simplicity.
By the Pieri rule, there is a unique GLpV q ˆ GLpW q-equivariant R-linear map
(4.1)

Sλp0q pV q b Sλp1q pW q b R Ð Sλp1q pV q b Sλp0q pW q b R.

Theorem 4.2. The bi-equivariant map (4.1) is injective.
We postpone the proof to §4.1 and now explain how it implies Theorem 4.1.
Proof of Theorem 4.1. Let |λp1q | ´ |λp0q | “ r. Choose a chain of partitions
λp1q “ αprq Ľ αpr´1q Ľ ¨ ¨ ¨ Ľ αp0q “ λp0q , with |αpiq | “ |λp0q | ` i for all i.
By Theorem 4.2, for i “ 1, . . . , r, there exists a sequence of bi-equivariant, linear injections
fi : Sαpiq pV q b Sαpi´1q pW q b R ãÑ Sαpi´1q pV q b Sαpiq pW q b R.
Let g be the composite map
F1 “ Sαprq pV q b Sαpr´1q pW q b ¨ ¨ ¨ b Sαp1q pW q b Sαp0q pW q b R
fr bidb¨¨¨bid



Sαprq pW q b Sαpr´1q pV q b ¨ ¨ ¨ b Sαp1q pW q b Sαp0q pW q b R
idbfr´1 b¨¨¨bid



..
.
idb¨¨¨bf2 bid



Sαprq pW q b Sαpr´1q pW q b ¨ ¨ ¨ b Sαp1q pV q b Sαp0q pW q b R
idb¨¨¨bidbf1



F0 “ Sαprq pW q b Sαpr´1q pW q b ¨ ¨ ¨ b Sαp1q pW q b Sαp0q pV q b R.
Clearly g is again injective. Since rankpF1 q “ rankpF0 q, we are done.



12

NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

4.1. Proof of Theorem 4.2. Now we prove Theorem 4.2. To cut down on indices, we write
λ for the smaller partition and µ for the larger. We put
µ “ pµ1 , . . . , µk q, and we assume µr ą µr`1 ,
λ “ µ except for λr “ µr ´ 1.
The proof relies on the Borel–Weil–Bott theorem and a construction of Eisenbud–Fløystad–
Weyman. We first review these results, then give an informal summary of the argument,
and finally give a proof of the theorem.
4.1.1. Borel–Weil–Bott and Eisenbud–Fløystad–Weyman. On PpW ˚ q, we have the short exact sequence
0 Ñ S Ñ W Ñ Op1q Ñ 0.
Note that we are using W , not W ˚ .
Given a permutation σ, define ℓpσq “ #ti ă j | σpiq ą σpjqu, the number of inversions.
Theorem 4.3 (Borel–Weil–Bott, [Wey03, Corollary 4.1.9]). Let β “ pβ1 , . . . , βk´1 q be weakly
decreasing and let d P Z. The cohomology of Sβ pSqpdq is determined as follows. Write
pd, β1 , . . . , βk´1 q ´ p0, 1, . . . , k ´ 1q “ pa1 , . . . , ak q.
(1) If ai “ aj for some i ‰ j, every cohomology group of Sβ pSqpdq vanishes.
(2) Otherwise, a unique permutation σ sorts the ai into decreasing order, aσp1q ą aσp2q ą
¨ ¨ ¨ ą aσpkq . Put λ “ paσp1q , . . . , aσpkq q ` p0, 1, . . . , k ´ 1q. Then
`
˘
Hℓpσq Sβ pSqpdq “ Sλ pW q,
and Hi pSβ pSqpdqq “ 0 for i ‰ ℓpσq.

We will also use the following result, on the existence of certain equivariant graded free
resolutions.
First, for a partition λ, we say pi, jq is an outer border square if pi, jq R λ and pi´1, j´1q P
λ (or i “ 1 or j “ 1), as in the ˚’s below, for λ “ p3, 1q:
˚ ˚ ˚ ˚ ¨¨¨
˚ ˚ ˚
˚ ˚
˚
˚
..
.

Let α be a partition with k parts, and let α1 Ľ α be obtained by adding at least one
border square in row 1, and all possible border squares in rows 2, . . . , k. Let αp0q “ α, and
for i “ 1, . . . , k, let αpiq be obtained by adding the chosen border squares only in rows 1, . . . , i.
Theorem 4.4 ([EFW11, Theorem 3.2]). Let E be a k-dimensional complex vector space
and R “ SympEq its symmetric algebra. There is a finite, GLpEq-equivariant R-module M
whose equivariant minimal free resolution is, with αpiq defined as above,
F0 Ð F1 Ð ¨ ¨ ¨ Ð Fk Ð 0,

Fi “ Sαpiq pEq b R.

Since the construction is equivariant, it works in families:
Theorem 4.5. Let X be a complex variety and E a rank k vector bundle over X. Let
E˚ Ñ X be the dual bundle. There is a sheaf M of OE˚ -modules with a locally-free resolution
F0 Ð F1 Ð ¨ ¨ ¨ Ð Fk Ð 0,

Fi “ Sαpiq pEq b OE˚ .

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

13

This follows by applying the EFW construction to the sheaf of algebras OE˚ “ SympEq.
The resolved sheaf M is locally given by M above. Note that M is coherent as an OX -module,
though we will not need this.
Remark 4.6. The construction we presented is also a direct corollary of a special case
Kostant’s version of the Borel–Weil–Bott theorem, for example see [ES17, §6] for some
discussion and references. We expect that other cases of Kostant’s theorem are relevant for
constructing complexes in the non-square matrix case.

4.1.2. Informal summary of the argument. We have fixed λ Ĺ µ, a pair of partitions differing
by a box. There is a unique Eisenbud–Fløystad–Weyman (EFW) complex with, in one step,
a linear differential of the form
Sµ pEq b SympEq Ñ Sλ pEq b SympEq.
The rest of the complex is uniquely determined by this pair of shapes, and is functorial in E.
We ‘sheafify’ the complex, lifting it to a complex of modules over the algebra SympV bOp´1qq
on PpW ˚ q, with terms of the form
Op´di q b Sα pV q b SympV b Op´1qq,
We twist so that the 0-th term has degree d “ µ1 , base change along the flat extension
SympV b Op´1qq ãÑ SympV b W ˚ q, and finally tensor through by the vector bundle Sβ pSq,
where S Ă W ˆ PpW ˚ q is the tautological rank-pk ´ 1q subbundle, and β is chosen so that
all the terms of the resulting complex except the desired pair have no cohomology. (That is,
Sβ pSq has supernatural cohomology with roots at each of the other di ’s.) Finally, we obtain
the desired map from the hypercohomology spectral sequence for the complex.
Example 4.7. Let k “ 4 and let λ “ p6, 1, 1, 0q, µ “ p6, 2, 1, 0q “

‹

(the added box

is starred). Working on PpW ˚q, the corresponding locally free resolution of sheaves (with
the twisting degrees indicated) is, after twisting and base-changing,
p6q Ð

p1q ÐÝ‹Ý

Ð

p´1q Ð

p´3q,

where αpdq stands for the sheaf Opdq b Sα pV q b SympV b W ˚ q on PpW ˚q. The desired linear
differential is marked with a ‹.
We put β “ p7, 1, 0q and tensor through by Sβ pSq (note that S has rank 3). Observe that
Sβ pSqpdq has no cohomology when d P t6, ´1, ´3u, but that
`
˘
`
˘
H1 Sβ pSqp1q “ S621 pW q,
H1 Sβ pSq “ S611 pW q.
Consequently, the E1 page of the hypercohomology spectral sequence only has the terms
S611 pV q b S621 pW q b R Ð S621 pV q b S611 pW q b R,
in the second and third columns. (The left term is along the main diagonal.) The spectral
sequence, run the other way, collapses with Hi pMq in the first column, where M is the sheaf
resolved by this complex. We see that the only nonvanishing term can be H0 pMq, giving an
exact sequence of R-modules
0 Ð H0 pMq Ð S611 pV q b S621 pW q b R Ð S621 pV q b S611 pW q b R Ð 0.



14

NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

‹

‚ ‚ ‚ ‚
‚
‚ ‚ ‚
‹
˝ ˝
˝
˝ ˝

Figure 4.1. Left: The partition µ; the starred box in the 4th row is removed to form
λ. Right: The outer strip is formed by connecting the inner border strip (‚) in rows 1 to
r ´ 1 to the outer border strip (˝) outside rows r ` 1, . . . , k. The empty squares form αp0q ;
then αpiq is obtained by adding all marked squares (‚, ˚, ˝) up to row i. Note that αp3q “ λ
and αp4q “ µ.

Remark 4.8. There are two easy ways to generalize the construction that we have sketched
above. First, in the map marked ‹ above, there is no reason to assume that the two partitions
differ by a single box, and the same construction allows them to differ by multiple boxes as
long as they are in the same row. In this case, the Pieri rule still implies that the map (4.1)
is unique up to scalar.
Second, in the above example we chose β so that Sβ pSqpdq has no cohomology for all d
besides the twists appearing in the target and domain of a single differential (in this case,
the one marked ‹). Alternatively, we could choose β so that Sβ pSqpdq has no cohomology for
all but two of the terms in the complex (not necessarily consecutive terms). The end result
is also a map of the form (4.1) where λp0q and λp1q differ by a connected border strip. In

general the map (4.1) is not unique up to scalar, however.
4.1.3. Combinatorial setup. We define shapes αpiq , i “ 0, . . . , k, as follows. Consider the
squares formed by:
‚ the inner border strip of µ inside rows 1, . . . , r ´ 1,
‚ the rightmost square in row r of µ,
‚ the outer border strip of µ outside rows r ` 1, . . . , k.
(See Figure 4.1.) Then α “ αp0q is obtained by deleting all these squares, and αpiq is
obtained by including those squares in rows 1, . . . , i. Clearly, αprq “ µ and αpr´1q “ λ.
Let ei be the number of border squares in row i, so
$
’
&1 ` µi ´ µi`1 for i “ 1, . . . , r ´ 1,
ei “ 1
for i “ r,
’
%1 ` λ ´ λ
for i “ r ` 1, . . . , k.
i´1
i
We define a modified (and negated) partial sum,
$
’
&µi`1 ´ i
di :“ µ1 ´ pe1 ` ¨ ¨ ¨ ` ei q “ µr ´ r
’
%µ ´ i ` 1
i

if i “ 0, . . . , r ´ 1,
if i “ r,
if i “ r ` 1, . . . , k.

Finally, we write β “ pβ1 , . . . , βk´1q for the unique partition such that, on PpW ˚ q, the
vector bundle Sβ pSqpdq has no cohomology for each di , i P t0, . . . , kuztr ´ 1, ru, where S Ă
W ˆ PpW ˚ q is the tautological rank-pk ´ 1q subbundle. By Borel–Weil–Bott, this determines
β uniquely by
β ´ p1, . . . , k ´ 1q “ pd0 , . . . , dr´2, dr`1 , . . . , dk q.

TOWARDS BOIJ–SÖDERBERG THEORY FOR GRASSMANNIANS

15

With these choices, we check:
Lemma 4.9. For d “ dr , the only nonvanishing cohomology of Sβ pSqpdq on PpW ˚ q is Hr´1 “
Sµ pW q. For d “ dr´1 , the only nonvanishing cohomology is Hr´1 “ Sλ pW q.
Proof. We apply Borel–Weil–Bott: we have to sort
pd, β1 , . . . , βk q ´ p0, 1, . . . , k ´ 1q “ pd, d0, . . . , dr´2 , dr`1, . . . , dk q.
For d “ dr´1 or dr , sorting takes r ´ 1 swaps, so in both cases Hr´1 is nonvanishing. To see
that the cohomology group is Sµ pW q for dr´1 and Sλ pW q for dr , we must check that
µ “ pd0 , . . . , dr´2, dr´1 , dr`1, . . . , dk q ` p0, 1, . . . , k ´ 1q,
λ “ pd0 , . . . , dr´2, dr , dr`1, . . . , dk q ` p0, 1, . . . , k ´ 1q
These are clear from the computation above.



4.1.4. The proof of Theorem 4.2. Let αpiq and di be defined as above. Consider the projective
space PpW ˚ q, with tautological line bundle Op´1q Ă W ˚ and rank-pk ´ 1q bundle S Ă W .
Set ξ :“ V b Op´1q. By Theorem 4.5, we have an exact complex
F0 Ð ¨ ¨ ¨ Ð Fi Ð Fi`1 Ð ¨ ¨ ¨ , where Fi “ Sαpiq pξq b Sympξq.
Note that
Sλ pξq “ Sλ pV q b Op´|λ|q.
For legibility, we write Op´λq for Op´|λ|q. Thus, we have a locally free resolution
Sαp0q pV q b Op´αp0q q b Sympξq Ð ¨ ¨ ¨ Ð Sαpiq pV q b Op´αpiq q b Sympξq Ð ¨ ¨ ¨
of sheaves of Sympξq-modules.
Next, let R “ OPpW ˚ q b SympV b W ˚ q. Observe that Sympξq ãÑ R is a flat ring extension
(locally it is an inclusion of polynomial rings). Now base change to R, which preserves
exactness. Finally, we tensor by Sβ pSq b Opαp0q ` µ1 q. Our final complex has terms
Sαpiq pV q b Sβ pSqpdi q b R.
Let M be the sheaf resolved by the complex.
We run the hypercohomology spectral sequence. Running the horizontal maps first, we
see that the sequence collapses on the E2 page with Hi pMq in the leftmost column. Running
the sequence the other way, the E1 page has terms
`
˘
Hq pSαppq pV q b Sβ pSqpdp q b Rq “ Sαppq pV q b Hq Sβ pSqpdp q b R.

(We emphasize that Sαppq pV q and R are trivial bundles.) By construction, the middle factor
is zero unless p “ r ´ 1, r, where by Lemma 4.9 the nonvanishing cohomology is Hr´1 , with
`
˘
`
˘
Hr´1 Sβ pSqpdr´1 q “ Sµ pW q,
Hr´1 Sβ pSqpdr q “ Sλ pW q.

In particular, the E1 page contains only the map

Sλ pV q b Sµ pW q b R Ð Sµ pV q b Sλ pW q b R,
with the left term located on the main diagonal. We see that Hi pMq “ 0 for i ą 0 and that
the map above is a resolution of H0 pMq by free R-modules.

16

NICOLAS FORD, JAKE LEVINSON, AND STEVEN V SAM

4.2. A stronger version of Theorem 4.1. Use the notation of Theorem 4.1. Since M is
torsion, the ranks must agree,
c0 Kλp0q pkq “ c1 Kλp1q pkq.
A straightforward choice of c0 , c1 is to take c0 “ Kλp1q pkq and c1 “ Kλp0q pkq, and to look for
a ‘small’ resolution of the form
(4.2)

M Ð Sλp0q pV q b Sλp1q pW q b R Ð Sλp1q pV q b Sλp0q pW q b R Ð 0,

equivariant for both GLpV q and GLpW q. Theorem 4.2 proves this conjecture when |λp1q | “
|λp0q | ` 1 and one can also do the case when λp1q is obtained by adding a connected border
strip to λp0q using Remark 4.8. In general, we do not know if this particular form of resolution
exists, though we conjecture that it does:
Conjecture 4.10. A small pure resolution (4.2) exists, for any pair λp0q Ĺ λp1q .
We finish by establishing one more situation where Conjecture 4.10 is true:
Proposition 4.11. Suppose there exists d so that pλp0q qi ď d ď pλp1q qj for all i, j. pEquivalently,
assume pλp0q q1 ď pλp1q qk .q Then a small pure resolution (4.2) exists.
Proof. We construct the map geometrically. After twisting down by d, we may suppose
instead λp0q ď 0 ď λp1q . Write λp0q “ ´µR for some partition µ ě 0. On X “ HompV, W q,
there is a canonical, bi-equivariant map of vector bundles T : V ˆ X Ñ W ˆ X, which is
an isomorphism away from the determinant locus. When λ ě 0, the Schur functor Sλ pV q is
functorial for linear transformations of V (when λ has negative parts, Sλ is only functorial
for isomorphisms), so there is an induced map
Sλp1q pTq : Sλp1q pV q ˆ X Ñ Sλp1q pW q ˆ X,
and, from the dual bundles, a second induced map
Sµ pT ˚ q : Sµ pW ˚q ˆ X Ñ Sµ pV ˚ q ˆ X.
Let g “ Sλp1q pTq b Sµ pT ˚ q. Note that g is generically an isomorphism of vector bundles, so
the corresponding map of R-modules is injective:
g : Sλp1q pV q b Sµ pW ˚ q b R Ñ Sµ pV ˚ q b Sλp1q pW q b R.
Finally, we note that there is a canonical isomorphism of representations Sµ pE ˚ q – S´µR pEq
for any vector space E. Apply this to the free R-modules above to get the desired map. 
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Mathematics Department, University of California, Berkeley, CA
E-mail address: njmford@gmail.com
URL: http://nicf.net
Mathematics Department, University of Michigan, Ann Arbor, MI
Current address: Mathematics Department, University of Washington, Seattle, WA
E-mail address: jlev@uw.edu
URL: http://math.washington.edu/~jlev/
Mathematics Department, University of Wisconsin, Madison, WI
E-mail address: svs@math.wisc.edu
URL: http://math.wisc.edu/~svs/

