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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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