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Minuscule Schubert varieties: poset polytopes, PBW-degenerated Demazure modules, and Kogan faces. (English) Zbl 1331.05225
Summary: We study a family of posets and the associated chain and order polytopes. We identify the order polytope as a maximal Kogan face in a Gelfand-Tsetlin polytope of a multiple of a fundamental weight. We show that the character of such a Kogan face equals to the character of a Demazure module which occurs in the irreducible representation of $$\mathfrak {sl}_{n+1}$$ having highest weight multiple of fundamental weight and for any such Demazure module there exists a corresponding poset and associated maximal Kogan face. We prove that the chain polytope parametrizes a monomial basis of the associated PBW-graded Demazure module and further, that the Demazure module is a favourable module, e.g. interesting geometric properties are governed by combinatorics of convex polytopes. Thus, we obtain for any minuscule Schubert variety a flat degeneration into a toric projective variety which is projectively normal and arithmetically Cohen-Macaulay. We provide a necessary and sufficient condition on the Weyl group element such that the toric variety associated to the chain polytope and the toric variety associated to the order polytope are isomorphic.

MSC:
 05E10 Combinatorial aspects of representation theory 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14M15 Grassmannians, Schubert varieties, flag manifolds 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 14N15 Classical problems, Schubert calculus
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 [1] Alexeev, V; Brion, M, Toric degenerations of spherical varieties, Selecta Math. (N.S.), 10, 453-478, (2004) · Zbl 1078.14075 [2] Ardila, F; Bliem, T; Salazar, D, Gelfand-tsetlin polytopes and feigin-Fourier-littelmann-Vinberg polytopes as marked poset polytopes, J. Combin. Theory Ser. A, 118, 2454-2462, (2011) · Zbl 1234.52009 [3] Backhaus, T., Bossinger, L., Desczyk, C., Fourier, G.: The degree of the Hilbert-Poincaré polynomial of PBW-graded modules. C. R. Math. Acad. Sci. Paris 352(12), 959-963 (2014) · Zbl 1333.17009 [4] Backhaus, T., Desczyk, C.: PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams. arXiv:1407.73664. J. Lie Theory 25(3), 815-856 (2015) · Zbl 1359.17018 [5] Cherednik, I., Feigin, E.: Extremal part of the PBW-filtration and E-polynomials. arXiv:1306.3146 (2013) · Zbl 0831.17004 [6] Cherednik, I., Orr, D.: Nonsymmetric difference Whittaker functions. arXiv:1302.4094 (2013) · Zbl 1372.20009 [7] De Loera, JA; McAllister, TB, Vertices of Gelfand-tsetlin polytopes, Discrete Comput. Geom., 32, 459-470, (2004) · Zbl 1057.05077 [8] Feigin, E; Fourier, G; Littelmann, P, PBW filtration and bases for irreducible modules in type an, Transform. Groups, 16, 71-89, (2011) · Zbl 1237.17011 [9] Feigin, E; Fourier, G; Littelmann, P, PBW filtration and bases for symplectic Lie algebras, Int. Math. Res. Not. IMRN, 1, 5760-5784, (2011) · Zbl 1233.17007 [10] Feigin, E., Fourier, G., Littelmann, P.: Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations. arXiv:1306.1292v3 (2013) · Zbl 06793800 [11] Feigin, E; Fourier, G; Littelmann, P, PBW-filtration over $$\mathbb{Z}$$ℤ and compatible bases for v(λ) in type an and cn, Springer Proceedings in Mathematics and Statistics, 40, 35-63, (2013) · Zbl 1323.17011 [12] Feigin, E., Makedonskyi, I.: Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration. Preprint arXiv:1407.6316 (2014) · Zbl 1319.05141 [13] Fourier, G.: New homogeneous ideals for current algebras: Filtrations, fusion products and Pieri rules, Preprint: arXiv:1403.4758. Moscow M. Journ. 15(1), 49-72 (2015) · Zbl 1383.17006 [14] Fourier, G.: PBW-degenerated Demazure modules and Schubert varieties for triangular elements. arXiv:1408.6939 (2014) · Zbl 1364.17009 [15] Gelfand, IM; Cetlin, ML, Finite-dimensional representations of the group of unimodular matrices, Doklady Akad. Nauk SSSR (N.S.), 71, 825-828, (1950) [16] Gonciulea, N; Lakshmibai, V, Degenerations of flag and Schubert varieties to toric varieties, Transform. Groups, 1, 215-248, (1996) · Zbl 0909.14028 [17] Hibi, T., Li, N.: Unimodular equivalence of order and chain polytopes. arXiv:1208.4029 (2012) · Zbl 1335.52026 [18] Kashiwara, M, The crystal base and littelmann’s refined Demazure character formula, Duke Math. J., 71, 839-858, (1993) · Zbl 0794.17008 [19] Kogan, M.: Schubert geometry of flag varieties and Gelfand-Cetlin theory. PhD-thesis (2000) · Zbl 0908.17010 [20] Kirichenko, VA; Smirnov, EYu; Timorin, VA, Schubert calculus and Gelfand-tsetlin polytopes, Uspekhi Mat. Nauk., 67, 89-128, (2012) [21] Littelmann, P, Crystal graphs and Young tableaux, J. Algebra, 175, 65-87, (1995) · Zbl 0831.17004 [22] Littelmann, P, Cones, crystals, and patterns, Transform. Groups, 3, 145-179, (1998) · Zbl 0908.17010 [23] Stanley, RP, Hilbert functions of graded algebras, Adv. Math., 28, 57-83, (1978) · Zbl 0384.13012 [24] Stanley, RP, Two poset polytopes, Discrete Comput. Geom., 1, 9-23, (1986) · Zbl 0595.52008
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