Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes

Schubert polynomials as projections of Minkowski sums of Gelfand-Tsetlin polytopes

Ricky Ini Liu Ricky Ini Liu, Department of Mathematics, North Carolina State University, Raleigh, NC 27695.
riliu@ncsu.edu
Karola Mészáros Karola Mészáros, Department of Mathematics, Cornell University, Ithaca, NY 14853 and School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540.
karola@math.cornell.edu
and  Avery St. Dizier Avery St. Dizier, Department of Mathematics, Cornell University, Ithaca NY 14853.
ajs624@cornell.edu
July 5, 2019
Abstract.

Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of . The integer point transform of the Gelfand-Tsetlin polytope projects to the Schur function . Schur functions form a distinguished basis of the ring of symmetric functions; they are also special cases of Schubert polynomials corresponding to Grassmannian permutations.

For any permutation with column-convex Rothe diagram, we construct a polytope whose integer point transform projects to the Schubert polynomial . Such a construction has been sought after at least since the construction of twisted cubes by Grossberg and Karshon in 1994, whose integer point transforms project to Schubert polynomials for all . However, twisted cubes are not honest polytopes; rather one can think of them as signed polytopal complexes. Our polytope is a convex polytope. We also show that is a Minkowski sum of Gelfand-Tsetlin polytopes of varying sizes. When the permutation is Grassmannian, the Gelfand-Tsetlin polytope is recovered. We conclude by showing that the Gelfand-Tsetlin polytope is a flow polytope.

Liu is partially supported by a National Science Foundation Grant (DMS 1758187). Mészáros is partially supported by a National Science Foundation Grant (DMS 1501059) as well as by a von Neumann Fellowship at the IAS funded by the Fund for Mathematics and the Friends of the Institute for Advanced Study.

1. Introduction

Schubert polynomials, introduced by Lascoux and Schützenberger in 1982 [15], are extensively studied in algebraic combinatorics [4, 6, 2, 7, 14, 23, 16, 19, 12, 11, 3]. They represent cohomology classes of Schubert cycles in flag varieties, and they generalize Schur functions, a distinguished basis of the ring of symmetric functions.

A well-known property of the Schur function is that it is a projection of the integer point transform of the Gelfand-Tsetlin polytope . This has inspired the following natural question for Schubert polynomials:

Question 1. For , is there a natural polytope and a projection map such that the projection of the integer point transform of under the map equals the Schubert polynomial ?

The construction of twisted cubes by Grossberg and Karshon in 1994 [8] is the first attempt at an answer to the above question. The integer point transforms of twisted cubes project to any Schubert polynomial. Indeed, Grossberg and Karshon show that for both flag and Schubert varieties, their (virtual) characters are projections of integer point transforms of twisted cubes. The one catch with twisted cubes is that they are not always honest polytopes; intuitively one can think of them as signed polytopal complexes. For the Grassmannian case they do not yield the Gelfand-Tsetlin polytope. Kiritchenko’s beautiful work [10] explains how to make certain corrections to the Grossberg-Karshon twisted cubes in order to obtain the Gelfand-Tsetlin polytope for Grassmannian permutations.

Recall that given a partition , the Gelfand-Tsetlin polytope is the set of all nonnegative triangular arrays

 x11 x12 ⋯ x1n x22 x23 ⋯ x2n ⋯ ⋯ xn−1,n−1 xn−1,n xnn

such that

 xin=λi for all 1≤i≤n, xi−1,j−1≥xij≥xi−1,j for all 1≤i≤j≤n.

To state our main result, which is a partial answer to Question 1, we need to consider the Minkowski sums of Gelfand-Tsetlin polytopes of partitions with different lengths.

Fix , and for each , let be a partition with parts (with empty parts allowed). We wish to study the Minkowski sum

 GT(λ(1))+GT(λ(2))+⋯+GT(λ(n)).

To make this Minkowski sum well-defined, we embed into for each . To do this, let be coordinates of and be coordinates of as in the definition of the Gelfand-Tsetlin polytope. The embedding is given by

 yij↦xi,j+n−k for all i+j≤k+1.

Given a column-convex diagram with rows, we associate to it a family of partitions in the following way. The shape , , has parts and is obtained from by ordering the columns of whose lowest box is in the th row in decreasing fashion and reading off according to the French notation. Note that is empty if there is no column of whose lowest box is in the th row.

Theorem 1.1.

The character of the flagged Schur module associated to a column-convex diagram with rows and is a projection of the integer point transform of

 (1) PD\coloneqqGT(λ(1))+GT(λ(2))+⋯+GT(λ(n))

with the embedding specified above. We obtain from the integer point transform via the specialization

 xij↦{x1when i=1,x−1i−1xiwhen i>1.

In the case that is the Rothe diagram of a permutation , the character of the flagged Schur module associated to is the Schubert polynomial . Thus, Theorem 1.1 answers Question 1 for permutations whose Rothe diagram is column-convex. The necessary background for and the proof of Theorem 1.1 is in Section 2. It is interesting to note that the Newton polytope of a Schubert polynomial is a generalized permutahedron [5, 20]; thus, the affine projection specified in Theorem 1.1 maps to a generalized permutahedron for column-convex .

Theorem 1.1 recovers Gelfand-Tsetlin polytopes for Grassmannian permutations. We conclude our paper by showing in Theorem 1.2 that Gelfand-Tsetlin polytopes are flow polytopes and by showing how to view in Theorem 1.1 in the context of flow polytopes.

Theorem 1.2.

is integrally equivalent to the flow polytope .

Section 3 contains the background for and the proof of Theorem 1.2, as well as some of its corollaries.

2. Polytopes projecting to Schubert polynomials

This section is devoted to proving Theorem 1.1 and explaining the relevant terminology. We start by defining diagrams, flagged Schur modules, and their characters.

2.1. Background

A diagram is a finite subset of . Its elements are called boxes. We will think of as a grid of boxes in matrix notation, so is the topmost and leftmost box. Canonically associated to each permutation is its Rothe diagram.

Definition 2.1.

The Rothe diagram of a permutation is the collection of boxes

 D(w)={(i,j)∣1≤i,j≤n,w(i)>j,w−1(j)>i}.

We can visualize as the set of boxes remaining in the grid after crossing out all boxes below or to the right of for each .

Definition 2.2.

A diagram is column-convex if for each , the set is an interval in .

Note that a Rothe diagram is column-convex if and only if avoids the patterns and .

Let be a diagram with rows. Denote by the symmetric group on the boxes in . Let be the subgroup of permuting the boxes of within each column, and define similarly for rows. Let denote the -vector space with basis indexed by fillings of . Observe that , , and act on on the right by permuting the filled boxes.

Define idempotents , in the group algebra by

 αD=1|Row(D)|∑w∈Row(D)w, βD=1|Col(D)|∑w∈Col(D)sgn(w)w,

where is the sign of the permutation . Given a filling , define to the be the linear combination

 eT=T⋅αDβD.

Identify with the tensor product , where and is the number of boxes of , in the following manner. First, fix an order on the boxes of . Then read each filling in this order to obtain a word on , and identify this word with the tensor , where is the standard basis of . As acts on , it acts diagonally on by acting on each component. This left action of on commutes with the right action of . Thus, the subspace of spanned by all elements is a submodule, called the Schur module of .

Call a filling of row-flagged if for all . Let be the subgroup of consisting of upper triangular matrices. The subspace of spanned by the elements for row-flagged forms a -submodule of , called the flagged Schur module of .

Definition 2.3.

The flagged Schur module of a diagram is the -submodule of spanned by

 {eT∣T is a row-flagged filling of D}.

The formal character , denoted by , is the polynomial

 sD=char(SD)(x1,…,xn)=Trace(X:SD→SD),

where is the diagonal matrix in with diagonal entries .

A particularly important subclass of characters of flagged Schur modules is that of Schubert polynomials as explained in Theorem 2.5 below. Schubert polynomials are associated to permutations, and they admit various combinatorial and algebraic definitions. For a permutation , we will define the Schubert polynomial via divided difference operators on polynomials.

Definition 2.4.

The Schubert polynomial of the long word for is defined as

 Sw0\coloneqqxn−11xn−22⋯xn−1.

For , there exists such that . For any such , the Schubert polynomial is defined by

 Sw\coloneqq∂iSwsi,

where

 ∂i(f)=f−sifxi−xi+1=f(x1,…,xn)−f(x1,…,xi−1,xi+1,xi,…,xn)xi−xi+1,

and is the transposition swapping and . The operators can be shown to satisfy the braid relations, so the Schubert polynomials are well-defined.

Schubert polynomials appear as the characters of flagged Schur modules of Rothe diagrams.

Theorem 2.5 ([13]).

Let be a permutation, be the Rothe diagram of , and be the character of the associated flagged Schur module . Then,

 Sw(x1,…,xn)=sD(w)(x1,…,xn).

2.2. Minkowski sums of Gelfand-Tsetlin polytopes

We now move towards proving Theorem 1.1, which for any column-convex diagram , relates the character with the Minkowski sum

 PD=GT(λ(1))+⋯+GT(λ(n))

defined in equation (1). To begin, we describe this Minkowski sum in terms of inequalities. We will need the following Lemma 2.6, which is proved in Section 3.

Lemma 2.6.

If has parts, then the Gelfand-Tsetlin polytope decomposes as a Minkowski sum:

 GT(λ)=n∑k=1(λk−λk+1)GT(1k0n−k).
Proposition 2.7.

Let be partitions such that has (possibly empty) parts. The Minkowski sum is defined by the following inequalities:

• for all , ; and

• for any positive integer and nonempty sequence of even length ,

 k∑s=1xjs−is,js−k−1∑s=1xjs+1−is,js+1≥ik∑s=0λ(n−s)j1−s,

with equality when and .

Remark 2.8.

A simple calculation shows that if, for instance, for some , then neither side of would change if we simply remove and from the sequence. Likewise, if for some , then neither side would change if we remove and from the sequence. Therefore we may equivalently take the inequalities for sequences .

One should observe that the entries occurring on the left side of lie at the corners of a path that zigzags southeast and southwest inside the triangular array.

Example 2.9.

Suppose . We first have inequalities and as with ordinary Gelfand-Tsetlin patterns. Then for , we get equalities

 x11=λ(3)1,x12=λ(2)1+λ(3)2,x13=λ(1)1+λ(2)2+λ(3)3,

as well as inequalities

 x22≥λ(3)2,x23≥λ(2)2+λ(3)3,andx33≥λ(3)3.

Finally, for , there is one more inequality, namely

 x12−x23+x33≥λ(3)2.
Proof of Proposition 2.7.

Let , and let be the polytope given by the inequalities above, . We first show that . For any point , choose, for each , points summing to it, so that . In particular, will contribute to a coordinate of the form if and only if .

Inequalities of the form are derived by summing the respective inequalities over all . For inequalities of type , consider a sequence , and suppose first that . Then

 k∑s=1y(n−m)js−is,js−m−k−1∑s=1y(n−m)js+1−is,js+1−m=y(n−m)j1−i1,j1−m+k−1∑s=1(y(n−m)js+1−is+1,js+1−m−y(n−m)js+1−is,js+1−m)≥λ(n−m)j1−m,

for each term in the sum is nonnegative by the defining inequalities of . If instead , then let be the minimum value such that . Then

 k′∑s=1y(n−m)js−is,js−m−k′∑s=1y(n−m)js+1−is,js+1−m =k′∑s=1(y(n−m)js−is,js−m−y(n−m)js+1−is,js+1−m)≥0

since again each term in the sum is nonnegative. Summing these inequalities over all then gives the desired inequality. In the case that and , we get equality since

 x1,j1=j1−1∑s=0y(n−s)1,j1−s=i1∑s=0λ(n−s)j1−s.

To show , we induct on and then the size of . First suppose . The inequalities involving are , and, when ,

 k−1∑s=1(xjs−is,js−xjs+1−is,js+1)+xjk,jk≥λ(n)j1=0

with equality if also and . These imply that for all and impose no additional constraints on the other entries. Removing the diagonal of entries then yields a triangular array that satisfies the inequalities defining . Therefore by induction

 Q(λ(1),…,λ(n−1),∅)=Q(λ(1),…,λ(n−1))=P(λ(1),…,λ(n−1))=P(λ(1),…,λ(n−1),∅).

If , then let be the number of nonzero parts. We will prove that , where we let for . This will prove the result by induction using Lemma 2.6 since then .

Recall that Gelfand-Tsetlin polytopes are integral polytopes. Given any integer point , set for , while for , set to be the minimum value such that and (if such an index exists, otherwise set ). Then define the point by if , otherwise .

We claim that . Our choice of guarantees that whenever , which ensures that for all . Therefore it suffices to show inequalities of type .

Given any sequence , suppose that for some , but . Consider what happens to the left hand side of if we insert between and , and we insert between and to get a new sequence . (Note that and .) This reduces the left hand side of by

 (x′j′−is−1,j′−x′js−is−1,js)−(x′j′−i′,j′−x′js−i′,js) =(x′js−1−is−1,js−1−x′js−is−1,js)−(x′tjs−1,js−1−x′tjs,js) =x′js−1−is−1,js−1−x′js−is−1,js ≥0,

while the right hand side of is unchanged. Thus for the sequence is implied by for the new sequence . Since , by iteratively applying this procedure to the new sequence, we will eventually arrive at a sequence for which such an does not exist.

It therefore suffices to prove inequality in the case that there exists some such that and exactly when . If , then the left hand side of is

 k∑s=1x′js−is,js−k−1∑s=1x′js+1−is,js+1 =(k∑s=1xjs−is,js−s′)−(k−1∑s=1xjs+1−is,js+1−s′+1) =k∑s=1xjs−is,js−k−1∑s=1xjs+1−is,js+1−1,

while the right hand side is

 μ(n)j1+ik∑s=1λ(n−s)j1−s=ik∑s=0λ(n−s)j1−s−1,

so this inequality follows from the corresponding inequality for . If , then consider the sequence obtained by inserting before , and after in the sequence. For , this yields the inequality

 (j1−1∑j=mxtj+1−1,j+k∑s=1xjs−is,js)−(j1−1∑j=mxtj+1,j+1+k−1∑s=1xjs+1−is,js+1)≥ik∑s=0λ(n−s)m−s.

But , and the right side is strictly greater than (since ). Thus

 k∑s=1xjs−is,js−k−1∑s=1xjs+1−is,js+1≥ik∑s=0λ(n−s)j1−s+1,

or equivalently,

 (k∑s=1xjs−is,js−s′)−(k−1∑s=1xjs+1−is,js+1−s′+1)≥ik∑s=0λ(n−s)j1−s=μ(n)j1+ik∑s=1λ(n−s)j1−s,

which is the inequality for . This completes the proof. ∎

2.3. Demazure operators and parapolytopes

To prove Theorem 1.1, we will need a formula for the character . The following formula is essentially a particular case of one due to Magyar [18]. (See also Reiner-Shimozono [22].) We first define the isobaric divided difference operator (or Demazure operator) acting on polynomials by

 πif(x1,…,xn)=∂i(xif)=xif−xi+1sifxi−xi+1,

where is the polynomial obtained from by switching and . Note that if is symmetric in and .

Proposition 2.10.

Let be a column-convex diagram with rows with . Define to be the diagram with rows such that , where . (Here, is obtained from by removing any column with a box in the first row and then shifting all remaining boxes up by one row.) Also let

 μ=(λ(1)1+λ(2)2+⋯+λ(n)n,λ(2)2+⋯+λ(n)n,…,λ(n)n),

the partition formed from all columns of with a box in the first row. Then

 sD=xμ11⋯xμnnπ1π2⋯πn−1(s˜D).
Proof.

Note that can be obtained from by switching the th and st row for , and then adding columns with boxes in rows for each . The result then follows immediately from [18] (see, for instance, Proposition 15). ∎

We now show that the polytope for can be constructed iteratively in a way that mimics the application of the operator . This geometric operation is the same as the operator given by Kiritchenko in [10] specialized for our current situation.

The key lemma is the following calculation.

Lemma 2.11.

Choose nonnegative integers , and , …, such that . Define the polynomial

 f(x1,x2)=ν1∑c1=μ1⋯νk∑ck=μkxN1−c1−⋯−ck1xc1+⋯+ck−N22.

Then

 π1f(x1,x2)=ν1∑c1=μ1⋯νk∑ck=μkνk+1∑ck+1=0xN1−c1−⋯−ck−ck+11xc1+⋯+ck+ck+1−N22,

where .

Proof.

Note that reversing the order of each of the summations in the expression for gives

 f=ν1∑c1=μ1⋯νk∑ck=μkxνk+1−N2+c1+⋯+ck1xN1−νk+1−c1−⋯−ck2=(x1x2)νk+1⋅s1f.

Hence

 π1f=x1f−x2s1fx1−x2=f⋅1−(x2x1)νk+1+11−x2x1=f⋅νk+1∑ck+1=0x−ck+11xck+12,

as desired. ∎

Consider with coordinates for . Let be the projection onto the coordinates for all .

Definition 2.12 ([10]).

A parapolytope is a convex polytope such that, for all , every fiber of the projection on is a coordinate parallelepiped.

In other words, for every and every set of constants (), there exist constants and (depending on the ) such that with for if and only if .

We denote this parallelepiped (which depends on and for ) by

 Π(μk,…,μn;νk,…,νn)=Π(μ,ν)={(xkj)nj=k∣μj≤xkj≤νj}⊂Rn+1−k.

Given a polytope , let be its integer point transform

 σP(xij)=∑(cij)∈P∩Z(n+12)∏1≤i≤j≤nxcijij,

and define to be the image of under the specialization sending

 xij↦{x1when i=1,x−1i−1xiwhen i>1.

In other words, the point corresponds to the monomial in which the exponent of is , where .

Lemma 2.13.

Fix , and let be parapolytopes. Suppose that for any fixed integer point , the fiber over of the projection on is the (integer) parallelepiped

 ΠP=Π(μk,…,μn−1,0;νk,…,νn−1,0),

while the fiber over of on is

 ΠQ=Π(μk,…,μn−1,μn;νk,…,νn−1,νn),

where and

 νn=n∑j=k−1ck−1,j+n∑j=k+1ck+1,j−n−1∑j=k(μj+νj)≥0.

Then .

Proof.

For fixed , the contribution to of the fiber over has the form

 <