# On The Tensor Semigroup Of Affine Kac-Moody Lie Algebras

Research paper by **Nicolas Ressayre**

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

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#### 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.