Multiplicities and tensor product coefficients for $A_r$

TL;DR

This paper introduces a fast Maple program for computing weight multiplicities and tensor product coefficients in $A_r$ Lie algebra representations using recent developments in vector partition functions, enhancing computational efficiency.

Contribution

It applies recent advances in vector partition functions to Kostant and Steinberg formulas, providing a new efficient computational tool for representation theory of $A_r$.

Findings

Developed a Maple program for $A_r$ multiplicities and tensor products.

The program computes locally polynomial functions for these multiplicities.

Enhanced computational speed and efficiency for representation calculations.

Abstract

We apply some recent developments of Baldoni-DeLoera-Vergne on vector partition functions, to Kostant and Steinberg formulas, in the case of $A_r$. We therefore get a fast {\sc Maple} program that computes for $A_r$: the multiplicity $c_{\lambda,\mu}$ of the weight $\mu$ in the representation $V(\lambda)$ of highest weight $\lambda$; the multiplicity $c_{\lambda,\mu,\nu}$ of the representation $V(\nu)$ in $V(\lambda)\otimes V(\mu)$. The computation also gives the locally polynomial functions $c_{\lambda,\mu}$ and $c_{\lambda,\mu,\nu}$.