A Demazure Character Formula for the Product Monomial Crystal

TL;DR

This paper establishes a Demazure character formula for certain truncations of the product monomial crystal associated with semisimple Lie algebras, linking it to generalized Demazure and Schur modules.

Contribution

It introduces a Demazure-type formula for the product monomial crystal and identifies it as the crystal of a generalized Demazure module and, in type A, as a generalized Schur module.

Findings

Truncations of the product monomial crystal are Demazure crystals.

A Demazure-type character formula for these crystals is provided.

In type A, the crystal corresponds to a generalized Schur module.

Abstract

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\mathbf{R}$.