Minuscule Schubert varieties: poset polytopes, PBW-degenerated demazure modules, and Kogan faces

Fourier, G. and Biswal, R. (2015) Minuscule Schubert varieties: poset polytopes, PBW-degenerated demazure modules, and Kogan faces. Algebras and Representation Theory, 18(6), pp. 1481-1503. (doi: 10.1007/s10468-015-9548-5)

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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 sln+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.

Item Type:Articles
Additional Information:The final publication is available at Springer via http://dx.doi.org/10.1007/s10468-015-9548-5
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Fourier, Dr Ghislain
Authors: Fourier, G., and Biswal, R.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Research Group:Algebra
Journal Name:Algebras and Representation Theory
Journal Abbr.:Algebr Represent Theor
Publisher:Springer Netherlands
ISSN:1386-923X
ISSN (Online):1572-9079
Copyright Holders:Copyright © 2015 Springer Netherlands
First Published:First published in Algebras and Representation Theory 18(6):1481-1503
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher.

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