Диссертация (1136188), страница 20

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Then Ow precedes Ow if and only if w = wsα for some root α and l(w ) = l(w) − 1 (see,e.g., [3, Theorem 2.11] or [15, Proposition 3.6.4]).For each root α, define the linear function (·, α) (that is, the coroot) on the weightlattice of G by the property sα λ = λ − (λ, α)α for all weights λ (the pairing (a, b) is often denoted by a, b∨ or by a, b). Denote by Z w the Schubert cycle represented by theclosure of the orbit Ow . The following result is proved in [3, Proposition 4.1] and [7,Proposition 4.4]:Hλ Z w =(λ, α)Z wsα ,αwhere the sum is taken over all positive roots α of G such that l(wsα ) = l(w) − 1. Inparticular, the coefficients (λ, α) are always nonnegative.One of our goals is to interpret this formula in terms of the Gelfand–Zetlin polytope Qλ . In what follows we will use the following equivalent formulation:Hλ Z w =α(wλ, α)Z sα w ,(2.1)Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012of O as a linear combination of the closures of Schubert cells at the boundary of O.

WeGelfand–Zetlin Polytopes and Flag Varieties2519where the sum is taken over all roots α such that w −1 α is positive and l(sα w) = l(w) −1.3 Correspondence between the Schubert Cells and the Faces of the Gelfand–ZetlinPolytopeIn this section, we will construct a correspondence between Schubert cycles and someof the faces of the Gelfand–Zetlin polytope Qλ (by Remark 2.1, it does not matter which3.1 Schubert cellsFix once and for all the diagonal maximal torus T ⊂ G. Everything below (weight vectors,Borel subalgebras, etc.) are assumed to be compatible with T. As before, we assume thatX is embedded into the projectivization P(Vλ ) of the irreducible representation Vλ as theG-orbit of the line spanned by a highest weight vector.We will use the following description of the Schubert cells from [3].

Let v ∈ Vλbe a non-zero weight vector with an extremal weight (recall that a weight is extremal ifit is of the form wλ for some element w in the Weyl group of G). Extremal weights areexactly the vertices of the weight polytope of Vλ , and their weight spaces are always onedimensional. In what follows, we will not distinguish between non-zero proportionalvectors with the same extremal weight. Let B be a Borel subgroup in G containing Tand b its Lie algebra. Note that all such Borel subgroups lie in the same orbit underthe action of the Weyl group W, and there are exactly |W| of them. Denote by U (b) theuniversal enveloping algebra of b.

Then the pair (v, B) defines the Schubert cell O(v, B),which is the B-orbit of v and the closure of this cell in the flag variety can be realized asfollows [3, Lemma 2.12]:O(v, B) = X ∩ P(U (b)v).Note that U (b)v is a B-invariant vector subspace in Vλ (called Demazure module). Itwould be natural to assign to the cell O(v, n) a face of the Gelfand–Zetlin polytope Qλby taking the convex hull of all basis vectors in the Gelfand–Zetlin basis that lie inthe subspace U (b)v (we identify the basis vectors in the Gelfand–Zetlin basis with theintegral points in Qλ ).

Unfortunately, it might happen that the convex hull is not a faceor has wrong dimension. However, this approach still works after some modification (seeSubsection 3.2).Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012weight λ we choose).2520V. KiritchenkoTwo Schubert cells O(v, B) and O(v , B ) are conjugate by the action of the Weylgroup (and hence represent the same cohomology class) if and only if B = w Bw −1 andv is proportional to wv for some element w ∈ W.

If we fix a Borel subgroup B, then theSchubert cells O(wv, B) for all w ∈ W give the full set of B-orbits in the flag variety. Inparticular,X = w∈W O(wv, B).For all possible choices of v and B, we get |W|2 Schubert cells forming |W| orbits underby sending (w1 , w2 ) to (ww1 , ww2 )). Note that there is no canonical identification (i.e.,independent of the choice of a Borel subgroup containing the torus T) between the cohomology classes of the cells O(v, B) and the elements of the Weyl group.

By differentchoices of a Borel subgroup, we assign to each cohomology class [O(v, B)] different elements of the Weyl group that are conjugate to each other. Since we use simultaneouslythe cells O(v, B) for different choices of B, we will not identify the Schubert cells withelements of the Weyl group.3.2 Faces of the Gelfand–Zetlin polytope.To each cell O(v, B) of dimension l, we now assign an l-dimensional face of the Gelfand–Zetlin polytope Qλ .

Recall that to each extremal vector v, there corresponds a uniquevertex of the Gelfand–Zetlin polytope, which we also denote by v. The vertex v is a uniquepreimage of a vertex of the weight polytope Pλ under the map p : Qλ → Pλ . It is easy toshow that the vertex v is simple [13, 2.2.2]. The edges coming out of v are in one-to-onecorrespondence with the roots α of G such that the root space gα does not annihilatethe extremal weight vector v [13, 2.2.3]. For each such root α denote by e(v, α) the edgecorresponding to α.

The edge e(v, α) is uniquely defined by the property that its projection under the map p : Qλ → Pλ is parallel to the root α (see Section 5 for more detailson simple vertices v and edges e(v, α)).Denote by R(v, B) the set of roots such that the root space gα ⊂ g is containedin b and does not annihilate v. The cardinality of R(v, B) is equal to the dimension ofthe cell O(v, B) [3, Lemma 2.2]. Let {β1 , . . . , βl } be all roots in the set R(v, B). Assign tothe Schubert cell O(B, v) the l-dimensional face (v, B) of the Gelfand–Zetlin polytopespanned by the edges e(v, β1 ), .

. . , e(v, βl ). There is a unique such face since the vertex vis simple. This face can be thought of as a lifting of the Demazure module U (b)v to theGelfand–Zetlin polytope.Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012the action of the Weyl group (so they can be identified with W × W where w ∈ W actsGelfand–Zetlin Polytopes and Flag Varieties2521Remark 3.1. There is an alternative description of the face (v, B) using the Morse theory on polytopes (such analog of the Morse theory was introduced in [12]).

Namely,choose a linear function fB on Rn−1 that takes positive values on all roots α whoseroot spaces gα are contained in b. Then the composition fB ◦ p is a linear function onthe Gelfand–Zetlin polytope. The face (v, B) is precisely the upper separatrix face forthe function fB ◦ p at the vertex v. The upper separatrix face is by definition the facespanned by all edges at v going upward with respect to the function fB ◦ p (that is,fB ◦ p increases along these edges).there is a Morse function on X given by the composition of the moment map X → Pλwith the same function fB , and the cells O(v, B) are the upper separatrix manifolds forthis Morse function (see [2, Section 4]).We now compare the Bruhat order on the cells O(v, B) with the inclusion orderon the faces (v, B).

It is easy to see that if (u, B) is a facet in (v, B), then O(u, B)precedes O(v, B) with respect to the Bruhat order. The converse is wrong, that is, ithappens already for G = SL 3 that O(u, B) lies at the boundary of O(v, B) in the flagvariety, but the face (u, B) does not belong to the face (v, B) (see Section 4). We saythat the face (v, B) is admissible if it contains all faces (u, B) such that the Schubertcell O(u, B) precedes the Schubert cell O(v, B) with respect to the Bruhat order.Denote by B − the lower triangular Borel subgroup, that is, B − is opposite to theBorel subgroup used to construct the Gelfand–Zetlin polytope.

If we fix B = B − and onlyvary v, then my correspondence between Schubert cycles and faces reduces to the correspondence defined in [13] (see Remark 5.1). Note that the collection of faces assignedto O(v, B − ) in [14, Section 4] always contains (v, B − ). In particular, if this collectionconsists of just one face (that is, of (v, B − )), then the corresponding Schubert cycle isa Kempf variety [13, Proposition 2.3.2], which is a very restrictive condition (see [13,Proposition 2.2.1] for a characterization of such Schubert cycles).

Equivalently, the corresponding Schubert polynomial consists of a single monomial. In particular, it is easyto check that in this case, (v, B − ) must be admissible. An advantage of my construction is that the freedom in the choice of B allows us to represent more general Schubertcycles by a single admissible face of the Gelfand–Zetlin polytope (see Remark 4.1).An interesting problem is to describe all admissible faces as well as the corresponding Schubert cycles. It is not true that for each Schubert cycle, there exists arepresentative O(v, B) such that the face (v, B) is admissible. There is a counterexample for the flag variety X4 of SL 4 (C).

Namely, if the closure of a Schubert cell O in X4Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012Note that the Bruhat cells O(v, B) can be defined in an analogous way. Namely,2522V. Kiritchenkois not smooth, then none of the faces corresponding to the cohomology class of O isadmissible (there are two such Schubert cells in X4 ). There is also one smooth Schubertcycle in SL 4 (C) which cannot be represented by an admissible face. For the flag varietyX3 of SL 3 (C), the closure of each Schubert cell is smooth, and every Schubert cycle canbe represented by an admissible face (see Section 4).We now state the Chevalley formula in terms of the Gelfand–Zetlin polytope.

Fora weight vector u, denote by p(u) the weight of u.α), then|( p(v), α)| = d(v, (u, B)),where d(v, (u, B)) is the integral distance from v to the face (u, B) as defined in Section2.2.This proposition will be proved in Section 5. If we apply it to formula (2.1), weimmediately get the following Chevalley formula for the admissible faces.Theorem 3.3. If the face (v, B) is admissible, then the Chevalley formula for the Schubert variety O(v, B) and the divisor Hλ can be written asHλ O(v, B) =d(v, (u, B))O(u, B),where the sum is taken over all Schubert cells O(u, B) that precede O(v, B).4 Example: Flag Variety for SL 3 (C)Figure 1 shows the Gelfand–Zetlin polytope Qλ for the irreducible representation ofSL 3 (C) with the highest weight λ = aω1 + bω2 .

This is a polytope in R3 (with coordinatesx, y, and z) defined by the following six inequalities:0 ≤ x ≤ a;a ≤ y ≤ b;x ≤ z ≤ y.The weight polytope Pλ is a hexagon in R2 . The polytope Qλ has six simple vertices whichare mapped bijectively to the vertices of the weight polytope Pλ under the map p. ThisDownloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012Proposition 3.2. If (u, B) is a facet of (v, B) (in particular, p(v) = sα p(u) for some rootGelfand–Zetlin Polytopes and Flag Varietiess1v2523vL1L2s2vs1s2vF1F2zys1s2s1vE2xs2s1vFig. 1. The Gelfand–Zetlin polytope Qλ for the irreducible representation of SL 3 (C).bijection is used to label the simple vertices of Qλ . Namely, we label by v the vertex thatgoes to the highest weight λ.

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