Abstract
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.
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Alexeev, V., Brion, M.: Toric degenerations of spherical varieties. Selecta Math. (N.S.) 10(4), 453–478 (2004)
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(8), 2454–2462 (2011)
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)
Backhaus, T., Desczyk, C.: PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams. arXiv:1407.73664. J. Lie Theory 25(3), 815–856 (2015)
Cherednik, I., Feigin, E.: Extremal part of the PBW-filtration and E-polynomials. arXiv:1306.3146 (2013)
Cherednik, I., Orr, D.: Nonsymmetric difference Whittaker functions. arXiv:1302.4094 (2013)
De Loera, J.A., McAllister, T.B.: Vertices of Gelfand-Tsetlin polytopes. Discrete Comput. Geom. 32(4), 459–470 (2004)
Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for irreducible modules in type A n . Transform. Groups 16(1), 71–89 (2011)
Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for symplectic Lie algebras. Int. Math. Res. Not. IMRN 1(24), 5760–5784 (2011)
Feigin, E., Fourier, G., Littelmann, P.: Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations. arXiv:1306.1292v3 (2013)
Feigin, E., Fourier, G., Littelmann, P.: PBW-filtration over \(\mathbb {Z}\) and compatible bases for v(λ) in type A n and C n . Springer Proceedings in Mathematics and Statistics 40, 35–63 (2013)
Feigin, E., Makedonskyi, I.: Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration. Preprint arXiv:1407.6316 (2014)
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)
Fourier, G.: PBW-degenerated Demazure modules and Schubert varieties for triangular elements. arXiv:1408.6939 (2014)
Gelfand, I.M., Cetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 71, 825–828 (1950)
Gonciulea, N., Lakshmibai, V.: Degenerations of flag and Schubert varieties to toric varieties. Transform. Groups 1(3), 215–248 (1996)
Hibi, T., Li, N.: Unimodular equivalence of order and chain polytopes. arXiv:1208.4029 (2012)
Kashiwara, M.: The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71(3), 839–858 (1993)
Kogan, M.: Schubert geometry of flag varieties and Gelfand-Cetlin theory. PhD-thesis (2000)
Kirichenko, V.A., Smirnov, E.Yu., Timorin, V.A.: Schubert calculus and Gelfand-Tsetlin polytopes. Uspekhi Mat. Nauk. 67(4(406)), 89–128 (2012)
Littelmann, P.: Crystal graphs and Young tableaux. J. Algebra 175(1), 65–87 (1995)
Littelmann, P.: Cones, crystals, and patterns. Transform. Groups 3(2), 145–179 (1998)
Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)
Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1(1), 9–23 (1986)
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Presented by Michel Brion.
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Biswal, R., Fourier, G. Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces. Algebr Represent Theor 18, 1481–1503 (2015). https://doi.org/10.1007/s10468-015-9548-5
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DOI: https://doi.org/10.1007/s10468-015-9548-5