Weight Multiplicity Polynomials of multi-variable Weyl Modules

TL;DR

This paper explores the polynomial dependence of weight space dimensions in multi-variable Weyl modules on the highest weight, providing explicit cases and analyzing the combinatorial structures involved.

Contribution

It proposes and supports a conjecture that weight space dimensions depend polynomially on the highest weight in multi-variable Weyl modules, with explicit examples and combinatorial insights.

Findings

Dimension of weight spaces depends polynomially on highest weight

Explicit calculations provided for up to three variables

Discussion of underlying combinatorial structures

Abstract

This paper is based on the observation that dimension of weight spaces of multi-variable Weyl modules depends polynomially on the highest weight (Conjecture 1). We support this conjecture by various explicit answers for up to three variable cases and discuss the underlying combinatorics.