On the tensor semigroup of affine kac-moody lie algebras

Abstract

In this paper, we are interested in the decomposition of the tensor product of two representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$. Let $P\_+$ be the set of dominant integral weights. For $\lambda \in P\_+$ , $L(\lambda )$ denotes the irreducible, integrable, highest weight representation of g with highest weight $\lambda$. Let $P\_+,\mathbb{Q}$ be the rational convex cone generated by $P\_+$. Consider the tensor cone $\Gamma (\mathfrak{g}):=\{(\lambda \_1,\lambda \_2,\mu )$\in$ P\_{+,\mathbb Q}^3\,| \exists N \textgreater{} 1 L(Nμ) \subset L(N λ\_1)\otimes L(N λ\_2)\}$. If $\mathfrak{g}$ is finite dimensional, $\Gamma (\mathfrak{g})$ is a polyhedral convex cone described in 2006 by Belkale-Kumar by an explicit finite list of inequalities. In general, $\Gamma (\mathfrak{g})$ is nor polyhedral, nor closed. In this article we describe the closure of $\Gamma (\mathfrak{g})$ by an explicit countable family of linear inequalities, when $\mathfrak{g}$ is untwisted affine. This solves a Brown-Kumar's conjecture in this case. We also obtain explicit saturation factors for the semigroup of triples $(\lambda \_1,\lambda \_2,\mu )$\in$P\_{+}^3$ such that $L(\mu )$\subset$L(\lambda \_1)\otimes L(\lambda \_2)$. Note that even the existence of such saturation factors is not obvious since the semigroup is not finitely generated. For example, in type $A,weprovethatanyinteger$d\geq 2$isasaturationfactor,generalizingthecase${\tilde A}\_1$ shown by Brown-Kumar.