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GELFAND­ZETLIN POLYTOPES AND DEMAZURE CHARACTERS
VALENTINA KIRITCHENKO, EVGENY SMIRNOV, AND VLADLEN TIMORIN

1. Introduction An important feature of toric geometry is the interplay between a polarized pro jective toric variety and its convex polytope. For instance, the Hilbert polynomial can be computed by counting integer points in the dilations of the polytope. In a recent preprint [KST], we explore the interplay between algebraic and convex geometry in a non-toric case, namely, for Schubert varieties in a complete flag variety. With a pro jective embedding of the flag variety, one can naturally associate a convex polytope, called the Gelfand­Zetlin polytope. In [Ko], Kogan assigned a collection of faces of the Gelfand­Zetlin polytope to each Schubert variety. Our main result is a formula for the Demazure character of a Schubert variety in terms of the exponential sums over the integer points in the union of these faces (Theorem 3.1). As a corollary, we get a formula for the Hilbert functions of Schubert varieties via the number of integer points (Corollary 3.2). This in turn implies a formula for the degrees of Schubert varieties via volumes (Corollary 3.3) similar to the Koushnirenko theorem in toric geometry. These results provide a generalization of [PS, Corollary 15.2] from Kempf varieties to all Schubert varieties. Denote by X the variety of complete flags in Cn , and by X w the Schubert variety corresponding to a permutation w Sn as in Section 2 (the codimension of X w is equal to the length l(w) of w). For every strictly dominant weight , denote by V the highest weight irreducible GLn -module with the highest weight . Recall that the Gelfand­Zetlin polytope P is a convex integer polytope in Rd , where d = n(n - 1)/2, with the property that the integer points inside and at the boundary of P parameterize a natural basis (Gelfand­Zetlin basis) in V (see Section 2 for a precise definition of P ). In particular, with each integer
The authors were supported by RFBR grant 10-01-00540-a, AG Laboratory NRU-HSE, MESRF grant, ag. 11.G34.31.0023, RF Innovation Agency grant 02.740.11.0608, Simons Foundation (ES,VT), Dynasty Foundation (VK), Deligne fellowship (VT), MESRF grants MK-2790.2011.1 (VT), 16.740.11.0307 (VK,ES), RFBR grants 11-01-00289-a (ES), 11-01-00654-a (VT), RFBR-CNRS grants 10-0193110-a (VK), 10-01-93111-a (ES).
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2 VALENTINA KIRITCHENKO, EVGENY SMIRNOV, AND VLADLEN TIMORIN

point z P we can associate its weight p(z ) in the character lattice of GLn . Denote by B - GLn (C) the subgroup of lower-triangular matrices. Consider the pro jective embedding X w X P(V ). Denote by - w () the Demazure character of the B - -module V,w := H 0 (X w , L ) where L is the restriction to X w P(V ) of the tautological line bundle on P(V ). For every and w, we prove that w () =
z A,w Zd

ep(z) ,

(1)

where A,w := w(F )=w F is the union of all rc-faces, or reduced Kogan faces F (see Section 2) of P with permutation w. - In the case w = e, that is, X w = X and V,w = V , formula (1) follows directly from the property of the Gelfand­Zetlin polytope mentioned above. For other w, it is usually not true that a subset of the Gelfand­ Zetlin basis (in particular, the subset given by the integer points in - A,w ) gives a basis in the Demazure module V,w . Our proof of formula (1) uses the Demazure character formula and elementary arguments involving combinatorics and geometry of the Gelfand­Zetlin polytope. In particular, we use a combinatorial procedure for dealing with divided difference operators (called mitosis) introduced in [KnM]. Our proof yields a geometric realization of mitosis [KST, Remark 6.7]. As a byproduct, we construct a realization of a simplex as a cubic complex different from those previously known [KST, Proposition 6.6]. 2. Definitions Denote by G the group GLn (C). The Weyl group of G is identified with the symmetric group Sn : a permutation w Sn corresponds to the element of G acting on the standard basis vectors ei by the formula ei ew(i) . For each w Sn , we define the Schubert variety X w to be the closure of the B - ­orbit of w in the flag variety X = G/B . It is easy to check that the length l(w) of w is equal to the codimension of X w in X . - Let V,w be the Demazure B - ­module defined as the dual space to the space of global sections H 0 (X w , L |X w ), where L is the very ample line bundle on X corresponding to a strictly dominant weight - . Note that by the Borel­Weil­Bott theorem V,e is isomorphic to the irreducible representation V of G with the highest weight . Choose - a basis of weight vectors in V,w . Recall that the Demazure character - w () of V,w is the sum over all weight vectors in the basis of the exponentials of the corresponding weights, or, equivalently, w () :=
µ

m

,w

(µ)eµ ,


GELFAND­ZETLIN POLYTOPES AND DEMAZURE CHARACTERS

3

Figure 1. A Gelfand­Zetlin polytope for GL3 where is the weight lattice of GLn and m,w (µ) is the multiplicity of - the weight µ in V,w . Let = (1 , . . . , n ) Zn be a strictly dominant weight of the group GLn (C), i.e. an n-tuple of integers i such that i < i+1 for all i = 1, . . . , n - 1. The Gelfand­Zetlin polytope P is a convex integer polytope in Rd , where d = n(n - 1)/2, defined by inequalities
1

2
1,1

1
,2

3

... ...
n-2,2 2,n-2 1,n-1



n



2,1

... n
-2,1

... ...
n-1,1

(GZ )

where (1,1 , . . . , 1,n-1 ; 2,1 , . . . , 2,n-2 ; . . . ; n ordinates in Rd , and the notation a c b

-2,1

,

n-2,2

;

n-1,1

) are co-

means a c b. See Figure 1 for a picture of the Gelfand­Zetlin polytope for G = GL3 . It will be convenient to represent faces of P by face diagrams. First, replace all j and i,j in table (GZ ) by dots. Every face of P is given by a system of equations of the form a = b, where a and b are coordinates represented by adjacent dots in two consecutive rows. To represent such an equation, we draw a line interval connecting the corresponding dots (these line intervals go from northeast to southwest or from northwest to southeast). Thus a system of equations defining a face of P gets represented by a collection of line intervals called the face


4 VALENTINA KIRITCHENKO, EVGENY SMIRNOV, AND VLADLEN TIMORIN

diagram. Rows of a face diagram are defined as the collections of dots corresponding to the coordinates i,j with a fixed i, and columns are by definition collections of dots with a fixed j (columns look like diagonals in our pictures). In what follows, we will consider faces of the Gelfand­Zetlin polytope given by the equations of the type i,j = i+1,j . We will call such faces Kogan faces. To each Kogan face F , we assign the permutation w(F ) as follows. First, assign to each equation i,j = i+1,j the elementary transposition si+j = (i + j, i + j + 1). Now compose all elementary transpositions corresponding to the equations defining F by going from left to right in each row of the diagram for F and by going from the bottom row to the top one. We say that a Kogan face F is reduced if the decomposition for w(F ) obtained this way is reduced. Reduced Kogan faces of the Gelfand­Zetlin polytopes are in bijective correspondence with reduced pipe-dreams (see [Ko, 2.2.1] for more details). Reduced Kogan faces for n = 3 (see Figure 1) are the vertex v , the edges E1 and E2 , the front faces F1 , F2 , the back face and P itself. The corresponding permutations are s2 s1 s2 , s2 s1 , s1 s2 , s2 , s2 , s1 , id, respectively. Note that the faces F1 and F2 have the same permutation. For each = (1 , . . . , n ), consider the affine hyperplane Rn-1 Rn with coordinates y1 , . . . , yn given by the equation y1 + . . . + yn + u0 = 0, where u0 = 1 + · · · + n . Choose coordinates u1 , . . . , un-1 in Rn-1 such that yi = ui - ui-1 for i = 1, . . . , n - 1. Consider the following linear map p : Rd Rn-1 from the space Rd with coordinates i,j to the hyperplane Rn-1 Rn :
n-i

ui =
j =1

i,j .

In other terms, if we arrange the coordinates i,j into a triangular table as in (GZ ), then ui is the sum of all elements in the i-th row. In what follows, we identify Rn with the real span of the weight lattice of G so that the i-th basis vector in Rn corresponds to the weight given by the i-th entry of the diagonal torus in G. Then the hyperplane Rn-1 is the parallel translate of the hyperplane spanned by the roots of G. It is easy to check that the image of the Gelfand­Zetlin polytope P Rd under the map p is the weight polytope of the representation V . Let S be a subset of the Gelfand­Zetlin polytope P (in what follows S will be a face or a union of faces). Define the character S of S as the sum of formal exponentials ep(z) over all integer points z S , that is, (S ) :=
z S Zd

ep(z) .

The formal exponentials eu , u Zn , generate the group algebra of . Thus the character takes values in this group algebra.


GELFAND­ZETLIN POLYTOPES AND DEMAZURE CHARACTERS

5

3. Results The main result of this section establishes a relation between the Demazure character of a Schubert variety and the character of the union of the corresponding faces. Theorem 3.1. For each permutation w Sn , the Demazure character w () is equal to the character of the union of al l Kogan faces in the Gelfand­Zetlin polytope P , whose permutation is w: w () =
w(F )=w

F .

If w is a 132­avoiding, or Kempf, permutation (such permutations are also called dominant ), then Theorem 3.1 reduces to [PS, Corollary 15.2]. Note that by [Ko, Proposition 2.3.2] a permutation w is Kempf if and only if there is a unique reduced Kogan face F such that w(F ) = w, and this is exactly the face considered in [PS]. Hence, w () = (F ) in this case. Let us now obtain several corollaries from this theorem. Firstly, we can easily describe the Hilbert function of the Schubert variety X w embedded into P(H 0 (X w , L |X w ) ) P(V ). Corollary 3.2. The dimension of the space H 0 (X w , L |X w ) is equal to the number of integer points in the union of al l reduced Kogan faces with permutation w: dim H 0 (X w , L |
X
w

)=
w(F )=w

F Zd .
X
w

In particular, the Hilbert function Hw, (k ) := dim H 0 (X w , Lk | equal to the Ehrhart polynomial of w(F )=w F , that is, H
w,

) is

(k ) =
w(F )=w

k F Z

d

(2)

for al l positive integers k . Secondly, we can compute the degree deg (X w ) of the Schubert variety X w in the embedding Xw P(V ). Denote by RF Rd the affine span of a face F . In the formulas displayed below, the volume form on RF is normalized so that the covolume of the lattice Zd RF in RF is equal to 1. Then Corollary 3.3. We have deg (X w ) = (d - l(w))!
w(F )=w

Volume(F )

(3)


6 VALENTINA KIRITCHENKO, EVGENY SMIRNOV, AND VLADLEN TIMORIN

Corollary 3.3 follows immediately from Corollary 3.2 by a standard argument from the theory of Newton polytopes [Kh], that is, by comparing the higher order terms in both sides of (2). Hence, (3) can be viewed as an asymptotic version of more precise identity (2). Note that in the general theory of Newton polytopes and Newton­Okounkov bodies developed recently by Kaveh and Khovanskii [KK] only asymptotic identities hold in most cases. So it is interesting that for Schubert varieties and corresponding unions of faces, we have an exact identity. References
[KK] Kiumars Kaveh, Askold Khovanskii, Newton convex bodies, semigroups of integral points, graded algebras and intersection theory, preprint arXiv:0904.3350v2 [Kh] Askold Khovanskii, Newton polyhedron, Hilbert polynomial, and sums of finite sets, Funct. Anal. and Appl., 26 (1992), no.4, 276­281 [KST] Valentina Kiritchenko, Evgeny Smirnov, Vladlen Timorin, Schubert calculus and Gelfand­Zetlin polytopes, preprint arXiv:1101.0278v1 [math.AG], submitted to Adv. Math. [KnM] Allen Knutson and Ezra Miller, Gr¨ obner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), 1245­1318 [Ko] Mikhail Kogan, Schubert geometry of flag varieties and Gelfand­Cetlin theory, Ph.D. thesis, Massachusetts Institute of Technology, 2000 [PS] Alexander Postnikov, Richard P. Stanley, Chains in the Bruhat order, J. Algebr. Comb. 29 (2009), no. 2, 133­174 E-mail address : vkiritchenko@yahoo.ca Laboratory of Algebraic Geometry and Faculty of Mathematics, Higher School of Economics, Vavilova St. 7, 112312 Moscow, Russia Institute for Information Transmission Problems, Moscow, Russia E-mail address : evgeny.smirnov@gmail.com Laboratory of Algebraic Geometry and Faculty of Mathematics, Higher School of Economics, Vavilova St. 7, 112312 Moscow, Russia Laboratoire J.-V. Poncelet (UMI 2615 du CNRS) and Independent University of Moscow, Moscow, Russia E-mail address : vtimorin@hse.ru Laboratory of Algebraic Geometry and Faculty of Mathematics, Higher School of Economics, Vavilova St. 7, 112312 Moscow, Russia Independent University of Moscow, Moscow, Russia