Computing weight $q$-multiplicities for the representations of the simple Lie algebras

TL;DR

This paper introduces a computational approach using SageMath to calculate $q$-multiplicities of weights in representations of simple Lie algebras, extending Kostant's classical formula with a $q$-analog.

Contribution

It develops a new SageMath implementation for computing $q$-multiplicities, providing a practical tool for representation theory research.

Findings

Successfully implemented a SageMath program for $q$-multiplicities

Enables efficient computation of weight multiplicities in simple Lie algebra representations

Extends Kostant's formula with a $q$-analog for deeper algebraic insights

Abstract

The multiplicity of a weight $\mu$ in an irreducible representation of a simple Lie algebra $\mathfrak{g}$ with highest weight $\lambda$ can be computed via the use of Kostant's weight multiplicity formula. This formula is an alternating sum over the Weyl group and involves the computation of a partition function. In this paper we consider a $q$-analog of Kostant's weight multiplicity and present a SageMath program to compute $q$-multiplicities for the simple Lie algebras.