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4, 119–146 (Russian). MR 1992080. English translation: Sb. Math. 194 (2003), no. 3–4,589–616.Department of Mathematics, Stony Brook University, Stony Brook NY 11794-3651,USAE-mail address: vkiritch@math.sunysb.eduПриложение C.Статья 3.Valentina Kiritchenko “Gelfand-Zetlin polytopes and flag varieties”International Mathematics Research Notices, Vol. 2010, No. 13, pp.2512–2531Разрешение на копирование: Согласноhttps://academic.oup.com/journals/pages/access_purchase/rights_and_permissions/self_archiving_policy_h автор статьи может использовать статью в образовательныхи научных целях внутри своего университета, если соблюдаются два условия:(1) использование некоммерческое, (2) указан источник.Kiritchenko, V.
(2010) “Gelfand–Zetlin Polytopes and Flag Varieties,”International Mathematics Research Notices, Vol. 2010, No. 13, pp. 2512–2531Advance Access publication December 29, 2009doi:10.1093/imrn/rnp223Gelfand–Zetlin Polytopes and Flag VarietiesValentina KiritchenkoCorrespondence to be sent to: valentina.kiritchenko@hcm.uni-bonn.deI construct a correspondence between the Schubert cycles on the variety of completeflags in Cn and some faces of the Gelfand–Zetlin polytope associated with the irreduciblerepresentation of SL n(C) with a strictly dominant highest weight. The construction ismotivated by the geometric presentation of Schubert cells using Demazure modules dueto Bernstein–Gelfand–Gelfand [3].
The correspondence between the Schubert cycles andfaces is then used to interpret the classical Chevalley formula in Schubert calculus interms of the Gelfand–Zetlin polytopes. The whole picture resembles the picture for toricvarieties and their polytopes.1 IntroductionLet G be the group SL n(C) and X = G/B the flag variety for G (here, B ⊂ G denotes a Borelsubgroup). The main goal of this paper is to translate to the flag variety some of the richinterplay that exists between geometry of toric varieties and combinatorics of convexpolytopes. As in the case of toric varieties, there is a polytope P H , namely Gelfand–Zetlinpolytope, naturally associated with each very ample divisor H on X. For a toric variety,an analogous polytope associated with a divisor H gives information about torus orbitsin the toric variety and their intersection products with H . For the flag variety, I willshow how to extract similar information about Schubert cycles in X and their intersection products with H using the Gelfand–Zetlin polytope P H .
In particular, the classicalReceived August 15, 2009; Revised November 14, 2009; Accepted November 24, 2009Communicated by Prof. Corrado De Concinic The Author 2009. Published by Oxford University Press. All rights reserved. For permissions,please e-mail: journals.permissions@oxfordjournals.org.Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012Institute for Information Transmission Problems, Bol’shoi Karetnyi per.19, Moscow, RussiaGelfand–Zetlin Polytopes and Flag Varieties2513Chevalley formula can be reformulated nicely in terms of Gelfand–Zetlin polytopes (seeTheorem 1.1 and Theorem 5.5). Toric and flag varieties are the most studied examples ofspherical varieties.
A further objective would be to use the relation between the geometry of flag varieties and Gelfand–Zetlin polytopes to get new insights into geometry ofmore general spherical varieties as outlined in [10].Recall that a Schubert or Bruhat cell is defined as an orbit of B in X under theleft action, and Schubert cycles are the cycles in the Chow ring of X represented by theclosures of Schubert cells.
Schubert cycles provide a basis in the Chow ring of X, andother hand, the cohomology ring of the flag variety is generated by the degree 2 classes(see, for instance [15, Theorem 3.6.15]). The group H 2 (X, Z) is isomorphic to the Picardgroup of X and can be identified with the weight lattice of G so that very ample divisors are identified with strictly dominant weights (see [4, 1.4.3]). Recall that the weightlattice of G is by definition the character lattice Zn−1 of a maximal torus in G. A centralformula in Schubert calculus is the Chevalley formula for the intersection product of aSchubert cycle with a divisor (see Subsection 2.3 for more details).
The Chevalley formula was proved independently by Bernstein–Gelfand–Gelfand [3] and Demazure [7] andwas already contained in a manuscript of Chevalley [6], which for many years remainedunpublished. This formula allows to express Schubert cycles in terms of divisors, thusrelating two different descriptions of the cohomology ring of the flag variety [3].Fix the upper triangular Borel subgroup B + . Let λ be a strictly dominant (withrespect to B + ) weight and Hλ the divisor corresponding to λ. We now assign to Hλ a convex polytope Qλ .
Recall that with each strictly dominant weight λ, one can associate theGelfand–Zetlin polytope Qλ (note that Zetlin is sometimes also transliterated as Cetlin orTsetlin). This is a convex polytope in Rd whose vertices lie in the integral lattice Zd ⊂ Rd(see Subsection 2.1 for the definition). Here, d = n(n− 1)/2 denotes the dimension of X.Let T be the diagonal maximal torus. The integral points inside and at the boundary ofQλ parameterize a natural basis of T-eigenvectors (introduced in [9]) in the irreduciblerepresentation Vλ of G with the highest weight λ.I will assign to each Schubert cycle in X a face of the Gelfand–Zetlin polytope(see Section 3). My construction depends on a choice of a Borel subgroup B containingthe maximal torus T (so in fact, I provide n! different correspondences between Schubertcycles and faces).
For each choice of B, we first construct a correspondence between Borbits and faces and then use the one-to-one correspondence between Schubert cyclesand B-orbits. The correspondence between B-orbits and faces preserves dimensions.The faces obtained for a given B correspond to Demazure B-modules in the representa-Downloaded from http://imrn.oxfordjournals.org/ at Higher School of Economics on February 29, 2012the latter is isomorphic to the cohomology ring H ∗ (X, Z) of X (see, e.g., [4, 1.3]). On the2514V.
Kiritchenkotion space Vλ . The freedom in the choice of a Borel subgroup allowed by this constructionis very useful. In many cases, it allows us to choose a face whose combinatorics capturesgeometry of a given Schubert cycle especially well (see Theorem 1.1 below). It might alsolead to an interesting realization of Schubert calculus in terms of Gelfand–Zetlin polytopes (this is work in progress with Evgeny Smirnov and Vladlen Timorin).
See Section 4for an example of such calculus in the case G = SL 3 (C).For a special choice of a Borel subgroup, namely for the lower triangular Borelsubgroup B − , my construction gives the correspondence between some of the Schubertmap X → Qλ [13] (see Section 3 for more details). In [14], Kogan and Miller extendedthis correspondence to all Schubert cycles: they assigned to each Schubert cycle a unionof faces using Caldero’s toric degenerations of flag varieties [5].
Both approaches (withmoment map and toric degenerations) work with B − -orbits, that is, there is only oneway to assign a face or a union of faces to a given Schubert cycle.For some of the faces of the Gelfand–Zetlin polytope that correspond to the Schubert cycles, the Chevalley formula for the intersection product of a Schubert cycle withthe divisor Hλ admits the following interpretation in terms of the respective face (cf.Theorem 3.3). We fix a Borel subgroup B containing T and, hence, fix a correspondencebetween Schubert cycles and faces. Denote by O the B-orbit corresponding to a face and by Z the Schubert cycle defined by O . In what follows, we only consider thosefaces that do correspond to Schubert cycles.
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