On products of sl_n characters and support containment

Abstract

Let $λ$, $μ$, $ν$ and $ρ$ be dominant weights of $\mathfrak{sl_n}$ satisfying $λ+ μ= ν+ ρ$. Let $V_λ$ denote the highest weight module corresponding to $λ$. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for $V_λ \otimes V_μ$ to be contained in $V_ν \otimes V_ρ$ as $\mathfrak{sl_n}$-modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible $\mathfrak{sl_n}$-module which appears in the decomposition of $V_λ \otimes V_μ$ does appear in the decomposition of $V_ν \otimes V_ρ$.

9 pages, 5 figures