Cyclic Demazure modules and positroid varieties

TL;DR

This paper explores the structure of positroid varieties through cyclic Demazure modules, establishing a canonical basis and crystal structure, thus advancing the understanding of their algebraic and combinatorial properties.

Contribution

It introduces the concept of cyclic Demazure modules, proves they have a canonical basis, and defines the cyclic Demazure crystal, linking algebraic and combinatorial aspects.

Findings

Cyclic Demazure modules have a canonical basis.

Defined the cyclic Demazure crystal.

Connected positroid varieties with Demazure modules.

Abstract

A positroid variety is an intersection of cyclically rotated Grassmannian Schubert varieties. Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which we call the cyclic Demazure module. In this note, we show that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.