Minuscule Schubert Varieties: Poset Polytopes, PBW-Degenerated Demazure Modules, and Kogan Faces

• Published in: Algebras and representation theory 2015-12, Vol.18 (6), p.1481-1503
• Main Authors: Biswal, Rekha, Fourier, Ghislain
• Format: Article
• Language: English
• Subjects: Associative Rings and Algebras; Commutative Rings and Algebras; Mathematics; Mathematics and Statistics; Non-associative Rings and Algebras
• ISSN: 1386-923X; 1572-9079
• DOI: 10.1007/s10468-015-9548-5

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 s l 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.