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On The Tensor Semigroup Of Affine Kac-Moody Lie Algebras

Research paper by Nicolas Ressayre

Indexed on: 09 Jan '17Published on: 09 Jan '17Published in: arXiv - Mathematics - Algebraic Geometry



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 $\mathfrak g$ with highest weight $\lambda$. Let $P\_{+,{\mathbb Q}}$ be the rational convex cone generated by $P\_+$. Consider the {\it tensor cone}$$\Gamma({\mathfrak g}):=\{(\lambda_1,\lambda_2,\mu)\in P_{+,{\mathbb Q}}^3\,|\,\exists N>1\quad L(N\mu)\subset L(N\lambda_1)\otimes L(N\lambda_2)\}.$$ 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 ofinequalities. 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 case $\tilde A\_n$, we prove that any integer $d\_0\geq 2$ is a saturation factor, generalizing the case $\tilde A\_1$ shown by Brown-Kumar.