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On products of sl_n characters and support containment

Research paper by Galyna Dobrovolska, Pavlo Pylyavskyy

Indexed on: 04 Aug '06Published on: 04 Aug '06Published in: Mathematics - Combinatorics



Abstract

Let $\lambda$, $\mu$, $\nu$ and $\rho$ be dominant weights of $\mathfrak{sl_n}$ 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 $\mathfrak{sl_n}$-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 $V_{\lambda} \otimes V_{\mu}$ does appear in the decomposition of $V_{\nu} \otimes V_{\rho}$.