Диссертация (1136188), страница 22
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If D(u) is obtained from D(v) by switching the edge ei,L j ,then the diagram D([u, v]) of the edge of the Gelfand–Zetlin polytope connecting u and vis obtained from D(v) by deleting the edge ei,L j .We now focus on the edges going out of a given simple vertex v. Their diagramsare obtained by deleting one of the edges of the diagram D(v). Denote by e the ith edgeDownloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012(see [13, 2.2.2]).
Namely, the diagram D(v) is simple if for all i = 2, . . . , n exactly n − i + 1Gelfand–Zetlin Polytopes and Flag Varieties2527of the tree T j for i = 1, . . . , j − 1 (that is the edge of the tree T j (v) starting at the ith rowof the diagram D(v) and ending at the (i + 1)-st row). Recall that we denoted by e(v, α)the edge of the Gelfand–Zetlin polytope whose projection p(e(v, α)) is parallel to the rootα.
It is easy to check using again the formula for the projection p : Qλ → Pλ (see Section2.1) that if we delete the edge e from the diagram D(v), we get the diagram of the edgee(v, α), where α = αi + αi+1 + . . . + α j−1 if e is of type L and α = −αi − αi+1 − . . . − α j−1 ife is of type R. Indeed, let pi,s and pi+1,s be the vertices of the edge e. Then switching eonly changes coordinates of v corresponding to the vertices of the tree T j (v) lying strictlystay the same for all other r. In particular, for each simple root αi , the diagram of theedge e(v, ±αi ) is obtained from D(v) by deleting the lowest edge (that is, the ith edge) ofthe tree Ti+1 (v), and the sign in ±α is determined by the slope of the lowest edge.
Thus,we get an explicit one-to-one correspondence between the edges e(v, α) of the Gelfand–Zetlin polytope and the edges of the diagram D(v).5.3 Faces (v, B) and proof of Proposition 3.2It is now easy to describe the diagrams of the faces (v, B) in terms of the diagram for v.Namely, we should delete all edges in D(v) that correspond to the roots in R(v, B) underthe above correspondence.Remark 5.1. If B = B − is the lower triangular, the diagram of (v, B − ) is obtained fromthe diagram D(v) by deleting all edges of type R.
The faces (v, B − ) are exactly the socalled Gelfand–Zetlin faces considered in [13, Subsection 2.2.1]. Note that the notationin [13] is different: my xi, j is his λi+ j,i and my σv is his wv−1 .We now determine under which conditions two simple vertices u and v of theGelfand–Zetlin polytope are connected by an edge.
The necessary condition p(u) = sα p(v)for some root α is obviously not sufficient (e.g., the vertices s2 v and s1 s2 v on Figure 1 arenot connected by the edge though p(s1 s2 v) = s1 p(s2 v)).Lemma 5.2. Let u and v be two simple vertices of the Gelfand–Zetlin polytope suchthat the weights p(u) and p(v) can be obtained from each other by the reflection sα withrespect to some root α.
Then u and v are connected by the edge if and only if the diagramDownloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012below pi,s . These coordinates increase by the same number xi−1,s+1 (v) − xi−1,s (v). Hence,the sums of coordinates n−rk=1 xr,k increase by the same number for r = i, . . . , j − 1 and2528V. KiritchenkoD(u) can be obtained from the diagram D(v) by switching the edge of D(u) correspondingto the root α.Proof. Choose α so that ( p(v), α) < ( p(u), α). Then the vertices v and u can only be connected by the edge e(v, α) (which will then coincide with the edge e(u, −α)), and thelemma immediately follows from the description of edges in the Gelfand–Zetlin polytope.Lemma 5.3.
If (u, B) is a facet of (v, B), then the vertices v and u are connected bythe edge.Note that for B = B − (as well as for B = B + ), the statement of the lemma isstraightforward. Indeed, by Remark 5.1, the diagrams of the faces (u, B − ) and (v, B − )are obtained from the diagrams of u and v, respectively, by deleting all edges of type R.On the other hand, the diagram D((v, B − )) is obtained from the diagram D((u, B − )) bydeleting one edge (since (u, B) is a facet of (v, B)). Hence, D(v) is obtained from D(u)by switching this edge. We now prove the lemma for arbitrary B.Proof. First, note that the assumptions of the lemma imply that O(u, B) precedesO(v, B) with respect to the Bruhat order.
Hence, p(u) = sα p(v) for some root α ∈ R(v, B).Rthe edge of the diagram D(v)Let (i j) be the transposition corresponding to sα and ei+1,scorresponding to the root α (we assume that this edge is of type R; type L case is completely analogous). We now compare the edges starting at the ith rows of the diagramsLbe the last edge of type L (when going from left to right)D(v) and D(u). Let ei+1,s−kRstarting from the ith row of the diagram D(v).
We want to show that k = 1 so that ei+1,scan be switched and the resulting diagram remains simple. Consider all edges of D(v)LRRRand ei+1,s, that is, the edges ei+1,s−k+1, . . . , ei+1,s−1. The above explicitbetween ei+1,s−kcorrespondence between simple vertices v and permutations σv implies that the treesTl (v) and Tl (u) have the same starting points unless l = i, j. From this, it is easy to deLLR, .
. . , ei+1,s. Moreover, if ei+1,s−k+lduce that the diagram D(u) contains the edges ei+1,s−k+1Lin D(v) for l = 1, . . . , k − 1 corresponds to a root β, then ei+1,s−k+l+1in D(u) correspondsLcorresponds to the root −α. Hence, the diagrams D((v, B)) andto −β. Finally, ei+1,s−k+1D((u, B)) will differ in at least k edges. Indeed, whenever the diagram D((v, B)) con-Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012To prove Proposition 3.2, we will need the following two lemmas.Gelfand–Zetlin Polytopes and Flag Varieties2529Rtains (or does not contain) the edge ei+1,s−k+l, the diagram D((u, B)) does not containLfor l = 1, . . .
, k − 1. Also, D((v, B)) does not contain(or contains) the edge ei+1,s−k+l+1L, while D((u, B)) does. It remains to note that the diagram of (v, B)the edge ei+1,s−k+1is obtained from the diagram of (u, n) by deleting exactly one edge (since (u, B) is afacet in (v, n)). Hence, k = 1.Lemma 5.4. If v and u are two simple vertices of the Gelfand–Zetlin polytope such thatp(v) = sα p(u) for the root α = αi + . . . + α j−1 , thenwhere s = σv−1 (i) and r = σv−1 ( j) (that is, p1,s and p1,r are the starting points of the treesTi (v) and T j (v), respectively).Proof. Since p(v) = σv λ, we have ( p(v), α) = (σv λ, α) = (λ, σv−1 α).
Note that the reflectiondefined by the root σv−1 α corresponds to the transposition (σv−1 (i) σv−1 ( j)) = (s r). Hence,|(λ, σv−1 α)| = |λr − λs |.We now prove Proposition 3.2. Let (u, B) be a facet of (v, B). By Lemma 5.3,the vertices v and u are connected by the edge. We have p(u) = sα p(v) for some root α.Suppose that α = αi + αi+1 + . .
. + α j−1 , where 0 < i < j < n. By Lemma 5.4, we have that|( p(v), α)| = |λr − λs |, where p1,s and p1,r are the starting points of the trees Ti (v) andT j (v), respectively. We now show that |λr − λs | = d(v, (u, B)). Denote by e the ith edgeof the tree T j (v). Since u and v are connected by the edge, we get by Lemma 5.2 thatthe diagram D(u) is obtained from D(v) by switching the edge e. We again consider thecase where e is of type R since the proof for the other case is completely the same. Letpi,l+1 and pi+1,l be the vertices of the edge e. Denote by F the facet of the Gelfand–Zetlinpolytope given by the equation xi−1,l = xi,l .
It is easy to check that (u, B) = F ∩ (v, B).Hence, the integral distance d(v, (u, B)) is by definition equal to the distance d(v, F ). Tocompute the latter, we note that the equation xi−1,l = xi,l defining F is already primitive.Since pi,l belongs to Ti (v) and pi+1,l to T j (v), we get that the xi−1,l coordinate of v is equalto λs and the xi,l coordinate to λr . Hence, d(v, F ) = λr − λs .5.4 Chevalley formula for arbitrary faces (v, B)The same arguments as in the proof of Proposition 3.2 allow us to prove a more generalChevalley-type formula for the faces of the Gelfand–Zetlin polytope. Let (v, B) be anyDownloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012|( p(v), α)| = |λr − λs |,2530V. Kiritchenko(not necessarily) admissible face and (u, B) a face such that O(u, B) precedes O(v, B)(but we no longer require that (u, B) ⊂ (v, B)).
Let α = αi + . . . + α j−1 be the root suchthat p(v) = sα p(u). Consider the edges e1 , . . . , ek that end at the (i + 1)-st row of thediagram D(u) and differ in the slope from the corresponding edges at the (i + 1)-st rowof D(v). Each such edge considered alone gives the diagram of a facet in the Gelfand–Zetlin polytope. Denote by Fi the facet defined by the edge ei .
Put d(v, u) := d(v, F1 ) +. . . + d(v, Fk).Hλ O(v, B) =d(v, u)O(u, B),where the sum is taken over all Schubert cells O(u, B) that precede O(v, B).Note that for admissible faces, Theorem 5.5 reduces to Theorem 3.3 (since wehave k = 1 by Lemma 5.3 and (u, n) = F1 ∩ (v, n)). Theorem 5.5 might be important fora realization of Schubert cycles by unions of faces of the Gelfand–Zetlin polytope [11].Proof.
The proof is almost the same as for admissible faces. We have |( p(v), α)| = |λr −λs | by Lemma 5.4. Assume that r > s. We can also write λr − λs as (λr − λik−1 ) + (λik−1 −λik−2 ) + . . . + (λi1 − λs ), where λi1 , . . . , λik−1 correspond to the starting points of the trees inD(u) containing the edges e1 , . . . , ek−1 , respectively.
It is easy to check that (λil − λil−1 ) =d(v, Fl ) using the same argument as in the proof of Proposition 3.2.AcknowledgmentsI am grateful to Michel Brion, Nicolas Perrin, and Evgeny Smirnov for the useful discussions. I amalso grateful to the referee for his comments. I would like to thank the Jacobs University Bremen,the Hausdorff Center for Mathematics, and the Max Planck Institute for Mathematics in Bonn forhospitality and support.References[1]Alexeev V., and M. Brion.
“Toric degenerations of spherical varieties.” Selecta Mathematica10 (2004): 453–78.[2]Atiyah, M. “Convexity and commuting Hamiltonians.” Bulletin of the London MathematicalSociety 14, no. 1 (1982): 1–15.Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012Theorem 5.5. Let (v, B) be any (not necessarily) admissible face.
ThenGelfand–Zetlin Polytopes and Flag Varieties[3]2531Bernstein, I.N., I.M. Gelfand, and S.I. Gelfand. “Schubert cells, and the cohomology of thespaces G/P .” Russian Mathematical Surveys 28 no. 3 (1973): 1–26.[4]Brion, M. “Lectures on the Geometry of Flag Varieties.” In Topics in Cohomological Studies[5]Caldero, P. “Toric degenerations of Schubert varieties.” Transformation Groups 7 no.
1 (2002),of Algebraic Varieties, 33–85. Trends in Mathematics. Basel: Birkhäuser, 2005.51–60.[6]Chevalley, C. “Sur les Décompositions Cellulaires des Espaces G/B.” In Algebraic Groupsand Their Generalizations: Classical Methods (University Park, PA, 1991), 1–23. Proceedingsof Symposia in Pure Mathematics 56. Providence, RI: American Mathematical Society, 1994.Demazure, M. “Désingularisation des variétés de Schubert généralisées.” Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday.
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