Vector partition function and representation theory

TL;DR

This paper leverages recent advances in vector partition functions to develop efficient Maple programs for calculating weight multiplicities, tensor product decompositions, and Ehrhart quasipolynomials in classical Lie algebras.

Contribution

It introduces new computational tools based on vector partition functions for classical Lie algebras, enabling efficient calculations of representation-theoretic quantities.

Findings

Developed Maple programs for Lie algebra computations.

Enabled calculation of weight multiplicities and tensor product coefficients.

Produced Ehrhart quasipolynomials related to Lie algebra representations.

Abstract

We apply some recent developments of Baldoni-Beck-Cochet-Vergne on vector partition function, to Kostant's and Steinberg's formulae, for classical Lie algebras $A\_r$, $B\_r$, $C\_r$, $D\_r$. We therefore get efficient {\tt Maple} programs that compute for these Lie algebras: the multiplicity of a weight in an irreducible finite-dimensional representation; the decomposition coefficients of the tensor product of two irreducible finite-dimensional representations. These programs can also calculate associated Ehrhart quasipolynomials.