Multiplicities and tensor product coefficients for $A_r$

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 $c_{\lambda ,\mu ,\nu }$.