On products of sl_n characters and support containment

Abstract

Let $\lambda$, $\mu$, $\nu$ and $\rho$ be dominant weights of $\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}}$ satisfying $\lambda +\mu =\nu +\rho$. Let $V_{\lambda }$ denote the highest weight module corresponding to $\lambda$. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for $V_{\lambda }\otimes V_{\mu }$ to be contained in $V_{\nu }\otimes V_{\rho }$ as $\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}}$-modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible $\mathfrak{s}{\mathfrak{l}}_{\mathfrak{n}}$-module which appears in the decomposition of $V_{\lambda }\otimes V_{\mu }$ does appear in the decomposition of $V_{\nu }\otimes V_{\rho }$.

9 pages, 5 figures