On the tensor semigroup of affine Kac-Moody lie algebras

Abstract:

The support of the tensor product decomposition of integrable irreducible highest weight representations of a symmetrizable Kac-Moody Lie algebra $\mathfrak {g}$ defines a semigroup of triples of weights. Namely, given $\lambda$ in the set $P_+$ of dominant integral weights, $V(\lambda )$ denotes the irreducible representation of $\mathfrak {g}$ with highest weight $\lambda$. We are interested in the tensor semigroup \begin{equation*} \Gamma _{\mathbb {N}}(\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+}^3\,|\, V(\mu )\subset V(\lambda _1)\otimes V(\lambda _2)\}, \end{equation*} and in the tensor cone $\Gamma (\mathfrak {g})$ it generates: \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+,{\mathbb {Q}}}^3\,|\,\exists N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}. \end{equation*} Here, $P_{+,{\mathbb {Q}}}$ denotes the rational convex cone generated by $P_+$.

In the special case when $\mathfrak {g}$ is a finite-dimensional semisimple Lie algebra, the tensor semigroup is known to be finitely generated and hence the tensor cone to be convex polyhedral. Moreover, the cone $\Gamma (\mathfrak {g})$ is described in Belkale and Kumar [Invent. Math. 166 (2006), pp. 185–228] by an explicit finite list of inequalities.

In general, $\Gamma (\mathfrak {g})$ is neither polyhedral, nor closed. In this article we describe the closure of $\Gamma (\mathfrak {g})$ by an explicit countable family of linear inequalities for any untwisted affine Lie algebra, which is the most important class of infinite-dimensional Kac-Moody algebra. This solves a Brown-Kumar’s conjecture in this case (see Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936]).

The difference between the tensor cone and the tensor semigroup is measured by the saturation factors. Namely, a positive integer $d$ is called a saturation factor, if $V(N\lambda _1)\otimes V(N\lambda _2)$ contains $V(N\mu )$ for some positive integer $N$ then $V(d\lambda _1)\otimes V(d\lambda _2)$ contains $V(d\mu )$, assuming that $\mu -\lambda _1-\lambda _2$ belongs to the root lattice. For $\mathfrak {g}={\mathfrak {sl}}_n$, the famous Knutson-Tao theorem asserts that $d=1$ is a saturation factor (see Knutson and Tao [J. Amer. Math. Soc. 12 (1999), pp. 1055–1090]). More generally, for any simple Lie algebra, explicit saturation factors are known. In the Kac-Moody case, $\Gamma _{\mathbb {N}}(\mathfrak {g})$ is not necessarily finitely generated and hence the existence of such a factor is unclear a priori. Here, we obtain explicit saturation factors for any affine Kac-Moody Lie algebra. For example, in type $\tilde A_n$, we prove that any integer $d\geq 2$ is a saturation factor, generalizing the case $\tilde A_1$ shown in Brown and Kumar [Math. Ann. 360 (2014), pp. 901–936].