Minuscule Schubert varieties: Poset polytopes, PBW-degenerated Demazure modules, and Kogan faces

TL;DR

This paper explores the combinatorial and geometric structures of minuscule Schubert varieties through poset polytopes, establishing connections with Demazure modules, Kogan faces, and toric degenerations, revealing new insights into their algebraic and geometric properties.

Contribution

It introduces a novel correspondence between poset polytopes and Demazure modules, demonstrating flat degenerations of Schubert varieties into toric varieties with desirable properties.

Findings

Order polytope as a maximal Kogan face in Gelfand-Tsetlin polytope

Chain polytope parametrizes a monomial basis of PBW-graded Demazure modules

Existence of conditions for toric variety isomorphism related to Weyl group elements

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 $\mathfrak{sl}_n$ 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.