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Let T̃ s denote the table, into which weput all coefficients of the polynomial h s , see Fig. 2. The lower left triangle of size s − 1 is the same inthe tables T s and T̃ s . The table T̃ s is skew-symmetric with respect to the main diagonal. These twoproperties give a unique characterization of the tables T̃ s .968P. Gusev et al. / Journal of Combinatorial Theory, Series A 120 (2013) 960–969Fig. 3. The rules of generating the tables T̃ s .The rules, by which the tables T̃ s are formed, are the following (see Fig.
3). The first table T̃ 1 isby definition the left-most table shown in Fig. 2. The next table T̃ s+1 is obtained inductively fromthe preceding table T̃ s in two steps. In the first step, we add to every element of T̃ s its immediatewest, south and southwest neighbors. In the second step, we modify elements in two diagonals ofthe table, namely, the elements, whose positions (measured by southwest corners) (k, m) satisfy theequality k + m = s or k + m = s + 2. To the cell at position (k, m), where k + m = s, we add the binomialk+mcoefficient m .
From the cell at position (k + 1, m + 1), we subtract this binomial coefficient.We have the following recurrence relation on the polynomials h s :h s+1 = h s (1 + x)(1 + z) + (1 − xz)(x + z)s ,which does not contain truncation operators. Therefore, the generating function H =fies the following linear equation:∞s=0 h s yssatis- −1 H = y (1 + x)(1 + z) H + (1 − xz) 1 − y (x + z).Solving this equation, we find thatH=y (1 − xz).(1 − y (x + z))(1 − y (1 + x)(1 + z))Knowing the generating function H , we can now obtain an explicit formula for the polynomials h s ,namely,h s (x, z) =1 − xz 1 + xz(1 + x)s (1 + z)s − (x + z)s .Theorem 1.4 is thus proved.Open problems.(1) Prove or disprove: the generating function G 4 is algebraic. Note that G 1 and G 2 are rational, andG 3 is algebraic.(2) Deduce differential or difference equations on the generating functions for the f -vectors and forthe modified h-vectors of Gelfand–Zetlin polytopes.AcknowledgmentsWe are grateful to the two anonymous referees for careful reading of the manuscript and extremelyhelpful suggestions.The authors were supported by RFBR grant 10-01-00540-a, AG Laboratory NRU-HSE, MESRF grant,ag.
11.G34.31.0023, the Simons-IUM fellowship (V.T.), Dynasty Foundation (V.K.), Deligne fellowship(V.T.), MESRF grants MK-2790.2011.1 (V.T.), MK-983.2013.1 (V.K.), RFBR grants 10-01-00739-a (V.T.),P. Gusev et al. / Journal of Combinatorial Theory, Series A 120 (2013) 960–96996911-01-00654-a (V.T.), 12-01-31429-mol-a (V.K.), 12-01-33020-mol-a-ved (V.T.), 12-01-33101-mol-aved (V.K.), RFBR-CNRS grant 10-01-93110-a (V.K.).This study comprises research findings from the “11-01-0159 Transformations between different geometric structures” and “12-01-0194 Algebraic cobordisms, polyhedral divisors and polytopes”projects carried out within The National Research University Higher School of Economics’ AcademicFund Program.References[1] V.
Guillemin, S. Sternberg, The Gel’fand–Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal. 52(1983) 106–128.[2] I.M. Gelfand, M.L. Cetlin, Finite dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk USSR(N.S.) 71 (1950) 825–828.[3] V. Kiritchenko, Gelfand–Zetlin polytopes and flag varieties, Int. Math. Res. Not. 13 (2010) 2512–2531.[4] V. Kiritchenko, E. Smirnov, V. Timorin, Schubert calculus and Gelfand–Zetlin polytopes, Russian Math. Surveys 67 (4) (2012)685–719.[5] M.
Kogan, E. Miller, Toric degeneration of Schubert varieties and Gelfand–Tsetlin polytopes, Adv. Math. 193 (1) (2005) 1–17.[6] J. Loera, T. McAllister, Vertices of Gelfand–Tsetlin polytopes, Discrete Comput. Geom. 32 (4) (2004) 459–470.[7] A. Okounkov, A remark on the Hilbert polynomial of a spherical manifold, Funktsional. Anal. i Prilozhen. 31 (2) (1997) 82–85(in Russian); translation in Funct.
Anal. Appl. 31 (2) (1997) 138–140.[8] A. Okounkov, Multiplicities and Newton polytopes, in: Kirillov’s Seminar on Representation Theory, in: Amer. Math. Soc.Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 231–244.Приложение H.Статья 8.Valentina Kiritchenko “Geometric mitosis”Mathematical Research Letters Volume 23, Number 4, 1071–1097, 2016Разрешение на копирование: Согласно Соглашению о копирайте автор статьиможет использовать статью в своих работах при условии, что указан источник.Math.
Res. Lett.Volume 23, Number 4, 1071–1097, 2016Geometric mitosisValentina KiritchenkoWe describe an elementary convex geometric algorithm for realizing Schubert cycles in complete flag varieties by unions of facesof polytopes. For GLn and Gelfand–Zetlin polytopes, combinatorics of this algorithm coincides with that of the mitosis on pipedreams introduced by Knutson and Miller. For Sp4 and a Newton–Okounkov polytope of the symplectic flag variety, the algorithmyields a new combinatorial rule that extends to Sp2n .1. IntroductionPositive presentations of Schubert cycles such as classical Schubert polynomials play a key role in the Schubert calculus.
Ideas of toric geometryand theory of Newton (or moment) polytopes motivated search for positivepresentations with a more convex geometric flavor. For instance, Schubertcycles on the complete flag variety for GLn were identified by various meanswith unions of faces of Gelfand–Zetlin polytopes [Ko, KoM, K10, KST]. Inthe present paper, we develop an algorithm for representing Schubert cycles by faces of convex polytopes in the case of complete flag varieties forarbitrary reductive groups.Let G be a connected reductive group, B ⊂ G a Borel subgroup, andX = G/B the complete flag variety.
There are several partially overlappingclasses of polytopes that can be associated with ample line bundles on X (seee.g. [BZ, L, GK, Ka13, K13]). For instance, string polytopes of Berenstein–Zelevinsky and Littelmann were recently exhibited in [Ka13] as Newton–Okounkov polytopes of flag varieties for a certain B-invariant valuation onX. These polytopes usually form families Pλ ⊂ Rd (where d := dim G/B)parameterized by dominant weights λ of G, and satisfy the property |Pλ ∩Zd | = dim Vλ where Vλ is the irreducible G-module with the highest weight2010 Mathematics Subject Classification: 14M15, 52B20, 05E10.Key words and phrases: Demazure operator, flag variety, Newton–Okounkovpolytope, Schubert calculus.This work was partially supported by the Russian Science Foundation grant 1421-00053 (Sections 3 and 4).10711072Valentina Kiritchenkoλ.
Moreover, there is a projection p : Rd → ΛG ⊗ R (where ΛG is the weightlattice of G) such that the Weyl character χ(Vλ ) can be expressed as themultiplicity free sum over the lattice points in Pλ :ep(x) .χ(Vλ ) =x∈Pλ ∩ZdRecall that ample line bundles Lλ on X are in bijective correspondencewith irreducible representations Vλ of G, and H 0 (X, Lλ ) = Vλ∗ . Let Xw be aSchubert variety, i.e., the closure of the B-orbit BwB/B in G/B, where w isan element of the Weyl group of G. Denote by χw (λ) := χ(H 0 (Xw , Lλ |Xw )∗ )the corresponding Demazure character.
The Demazure character χw (kλ) fork ∈ N can be thought of as a refined version of the Hilbert polynomial of theprojective embedding Xw ⊂ P(Vλ ), in particular, it can be used to computethe degree deg(X w ) of Xw in P(Vλ ). A natural way to identify Xw with aunion of faces Γ∈Sw Γ := Sw ⊂ Pλ is to choose the set Sw of faces so thatthe following identity holds for all λ (cf. [KST, Theorem 5.1] for G = GLn ):χw (λ) =ep(x) .x∈Sw ∩ZdIn particular, dim H 0 (Xw , Lλ |Xw ) is equal to |Sw ∩ Zd |, which yields theidentity in the spirit of the theory of Newton–Okounkov convex bodies (cf.[KST, Theorem 5.4] for G = GLn ):Volume(Γ).deg(Xw ) = dim(Xw )!Γ∈SwBy the Demazure character formula, Demazure characters can be calculated inductively starting from the class of a point Xid = {pt} (that is,χid (λ) = eλ ), and applying Demazure operators D1 , .
. . , Dr corresponding tothe simple roots of G. In this paper, we define geometric mitosis operationsM1 , . . . , Mr on faces of Pλ as convex geometric counterparts of Demazureoperators D1 , . . . , Dr , that is, they satisfy the identity⎛⎞ep(x) ⎠ =ep(x) .Di ⎝x∈Sw ∩Zdx∈Mi (Sw )∩Zdwhenever l(si w) = l(w) + 1 (see Theorem 3.4).The definition of mitosis operations is elementary, and its main ingredient is mitosis on parallelepipeds introduced in [KST, Section 6]. MitosisGeometric mitosis1073assigns to every face a collection (possibly empty) of faces of dimensionone greater. Mitosis on parallelepipeds can be viewed as a convex geometricrealization of the mitosis of Knutson–Miller [KnM, M] restricted to two consecutive rows of pipe dreams (see Section 2.1 for a reminder on pipe dreams).We use mitosis on parallelepipeds as a building block for mitosis on moregeneral polytopes Pλ (called parapolytopes) that admit r different fibrationsby parallelepipeds.
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