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(N\mu) \subset L(N \lambda\_1)\otimes L(N \lambda\_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 , we prove that any integer $d\geq 2$ is a saturation factor, generalizing the case ${\tilde A}\_1$ shown by Brown-Kumar.