Диссертация (1136188), страница 19

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We say that a face is admissible if for eachcodimension one orbit O in the closure of the orbit O , the face contains the face .In other words, the Bruhat order on Schubert cycles agrees with the natural order onfaces given by inclusion.Theorem 1.1. For any admissible face we haveHλ Z =d(v, )Z ,⊂where the sum is taken over the facets of (that correspond to the Schubert cellsO of codimension 1 at the boundary of O ). Here, v is a fixed vertex of the face and d(v, ) denotes the integral distance from v to the face (see Section 2.2 for thedefinition).Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012cells and faces of the Gelfand–Zetlin polytope constructed by Kogan using the momentGelfand–Zetlin Polytopes and Flag Varieties2515Note that in this form, the Chevalley formula is completely analogous to thewell-known formula for toric varieties (e.g., see [8, Section 3.4] or [10]).

There is a generalization of Theorem 1.1 that holds for all faces (see Theorem 5.5).Many Schubert cycles can be represented by an admissible face for differentchoices of B but not all of them, for example, for G = SL 3 , all Schubert cycles can berepresented by admissible faces (and all Schubert cycles are smooth). For G = SL 4 , thereare two non-smooth Schubert cycles, and none of them can be represented by an admissible face. There might be some relation between smoothness of Schubert cycles andthen already for SL 3 there will be a Schubert cycle such that the corresponding face isnot admissible (see Remark 4.1).It might be possible to extend the correspondence between Schubert cycles andfaces constructed in this paper to the complete flag varieties for other reductive groupsby replacing the Gelfand–Zetlin polytope with appropriate string polytopes (see [1, Definition 1.2] for a definition of string polytopes).This paper is organized as follows. In Section 2, we recall the definition of theGelfand–Zetlin polytope and the notion of integral distance.

We also state the classical Chevalley formula. Section 3 contains the main results: the construction of correspondences between faces of the Gelfand–Zetlin polytope and Schubert cycles and theChevalley formula in terms of the Gelfand–Zetlin polytope (Theorem 3.3). In Section 4,we consider in detail the example G = SL 3 . In Section 5, we study combinatorics andgeometry of the Gelfand–Zetlin polytope and prove Theorem 3.3. We also formulate andprove an extension of Theorem 3.3 to non-admissible faces (Theorem 5.5).2 Gelfand–Zetlin Polytopes and Chevalley FormulaIn this section, we recall the definition of the Gelfand–Zetlin polytope and the Chevalleyformula for the intersection product of a Schubert cycle with a divisor.

We also discussthe notion of integral distance.2.1 Gelfand–Zetlin polytopeLet λ = (λ1 , . . . , λn) be a strictly increasing collection of n integer numbers. To each suchcollection, we assign the irreducible representationπλ : G → G L(Vλ )Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012their representability by admissible faces. Note that if we only take B − (as in [13, 14]),2516V. Kiritchenkowith the strictly dominant highest weight (λ2 − λ1 )ω1 + . . .

+ (λn − λn−1 )ωn−1 (which willalso be denoted by λ), where ω1 , . . . , ωn−1 are the fundamental weights of G. To definethe fundamental weights, we fix the diagonal maximal torus T and the upper triangular Borel subgroup B + . The Gelfand–Zetlin polytope Qλ associated with λ is a convexpolytope in Rd (recall that d = n(n− 1)/2) defined by the inequalitiesλ1λ2λ3...x1,2...x2,1...λnx1,n−1x2,n−2...xn−2,1xn−2,2xn−1,1where (x1,1 , . . .

, x1,n−1 ; x2,1 , . . . , x2,n−2 ; . . . ; xn−2,1 , xn−2,2 ; xn−1,1 ) are coordinates in Rd andthe notationabcmeans a ≤ c ≤ b. See Figure 1 for a picture of the Gelfand–Zetlin polytope for G = SL 3 .There is a T-eigenbasis in Vλ such that its vectors are in one-to-one correspondence with the integral points inside Qλ (see for instance [14, Section 5] for the description of this basis). We will denote by the same letter v an integral point in Qλ and thecorresponding basis vector in Vλ . There is a natural map p that assigns to each integralpoint v the weight of the corresponding basis vector v ∈ Vλ . Let us extend this map bylinearity to the map p : Rd → Rn−1 . Denote by Pλ ⊂ Rn−1 the weight polytope of the representation Vλ .

The map p sends the Gelfand–Zetlin polytope Qλ to the weight polytopePλ and can be written in coordinates as follows [13, 2.1.2]. Let α1 , . . . , αn−1 be the simpleroots of G (so they form a basis in Rn−1 dual with respect to the Cartan–Killing form tothe basis of the fundamental weights ω1 , . . .

, ωn−1 ). Then we havep : (xi j ) → (n−1i=1x1,i )α1 +(n−2x2,i )α2 + . . . + (xn−2,1 + xn−2,2 )αn−2 + xn−1,1 αn−1i=1+ constant vector.Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012x1,1...Gelfand–Zetlin Polytopes and Flag Varieties2517Remark 2.1.

Note that for any two strictly dominant weights λ and μ, the corresponding Gelfand–Zetlin polytopes Qλ and Qμ are analogous, that is, have the same normalfan. In particular, there is a bijective correspondence between their faces. This is similar to the toric case, where polytopes corresponding to any two very ample divisors areanalogous.2.2 Integral distancethe affine space Rd.

Let H be a hyperplane spanned by lattice vectors and v ∈ Zd an integral point. Then the integral distance d(v, H ) from v to the hyperplane H is the index inZd of the subgroup spanned by the vectors v − u for all u ∈ H . By definition, the integraldistance is invariant under unimodular linear transformations of Rd. To compute theintegral distance, we first find a primitive integral equation f(x) = 0 defining H , that is,f(x) = a0 + a1 x1 + .

. . + adxd where ai ∈ Z and the greatest common divisor of a0 , . . . , adis 1. It is then easy to check that the integral distance between v and H is equal to theabsolute value of f(v). In particular, if H is parallel to a coordinate hyperplane, then theintegral distance coincides with the Euclidean distance.Example 2.2. In what follows, we will be interested in the case where H is a hyperplanecontaining a facet of the Gelfand–Zetlin polytope Qλ . A primitive integral equation for His then either xi, j − xi−1, j = 0 or xi, j − xi−1, j+1 = 0 for some i = 1, . .

. , n − 1, j = 1, . . . , n −i (we put x0, j = λ j ). Hence, the integral distance from a point v = (vi, j ) to H is equal tovi, j − vi−1, j or to vi−1, j+1 − vi, j , respectively.In the sequel, we will use the notion of integral distance in the following setting.Let P be a convex lattice polytope of dimension d in Rd. Recall that a vertex u of P iscalled simple if exactly d facets intersect in u (or equivalently, exactly d edges meet atu).

In other words, in the neighborhood of u, the polytope P looks like a d-dimensionalsimplex. Let ⊂ P be a face of P , and ⊂ a facet of that contains at least onesimple vertex of P . This ensures that there is a unique hyperplane H such that H ∩ P isa facet of P and H ∩ = . For any integral point v ∈ , we can now define the integraldistance d(v, ) as the integral distance from v to the hyperplane H . Such distances arisenaturally in toric geometry when one computes products of toric orbits with divisors.Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012Below, we recall the notion of integral distance.

Consider the integral lattice Zd ⊂ Rd in2518V. Kiritchenko2.3 Bruhat order and Chevalley formulaFix a strictly dominant weight λ. Recall that Vλ denotes the irreducible representationwith the highest weight λ. We assume that X = G/B is embedded into the projectivespace P(Vλ ) as the G-orbit of the line spanned by a highest weight vector v ∈ Vλ . Denoteby Hλ the divisor of hyperplane section on X (this is one of the equivalent ways to identify strictly dominant weights with very ample divisors [4, 1.4]).

For each Schubert cell Oin X, the Chevalley formula computes explicitly the intersection of Hλ with the closurewill now state this formula.First, recall that the choice of a Borel subgroup B in G defines a one-to-one correspondence between the Schubert cells in X and elements of the Weyl group W of G.We identify the Weyl group with N(T)/T, where N(T) is the normalizer of T in G. Thenthe Schubert cell Ow is the B-orbit of the line spanned by wv ∈ Vλ . Note that the lengthl(w) (defined as the minimal number of simple reflections in a decomposition of w) isequal to the dimension of Ow .

Recall that there is a natural partial order on Schubertcells called Bruhat order. We say that Ow precedes Ow with respect to the Bruhat orderif Ow is contained in the closure of Ow and dim Ow = dim Ow − 1. In other words, Owis a boundary divisor in Ow . The Bruhat order can also be defined in terms of the Weylgroup as follows. Denote by sα the reflection in the hyperplane perpendicular to a rootα.

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