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].

---

中文翻译：

---

关于仿射Kac-Moody李代数的张量半群

摘要：支持对称Kac-Moody李代数$\mathfrak {g}$的可积不可约最高权重表示的张量积分解定义了权重三元组的半群。即，给定主积分权重集合 $P_+$ 中的 $\lambda$，$V(\lambda )$ 表示具有最高权重 $\lambda$ 的 $\mathfrak {g}$ 的不可约表示。我们感兴趣的是张量半群\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*} 和张量锥$\Gamma (\mathfrak {g})$ 它生成： \begin{equation*} \Gamma (\mathfrak {g})≔\{(\lambda _1,\lambda _2,\mu )\in P_{+, {\mathbb {Q}}}^3\,|\,\存在 N\geq 1 \quad V(N\mu )\subset V(N\lambda _1)\otimes V(N\lambda _2)\}。\end{equation*} 这里，$P_{+,{\mathbb {Q}}}$ 表示由 $P_+$ 生成的有理凸锥。在 $\mathfrak {g}$ 是有限维半单李代数的特殊情况下，已知张量半群是有限生成的，因此张量锥是凸多面体。此外，锥 $\Gamma (\mathfrak {g})$ 在 Belkale 和 Kumar [Invent. 数学。166 (2006), pp. 185–228] 通过显式的有限不等式列表。一般来说，$\Gamma (\mathfrak {g})$ 既不是多面体，也不是封闭的。在这篇文章中，我们描述了 $\Gamma (\mathfrak {g})$ 的闭包，对于任何非扭曲仿射李代数，它是最重要的无限维 Kac-Moody 代数类的线性不等式的可数族。在这种情况下，这解决了 Brown-Kumar 猜想（参见 Brown 和 Kumar [Math. Ann. 360 (2014), pp. 901–936]）。张量锥和张量半群之间的差异由饱和因素。即正整数$d$称为饱和因子, 如果 $V(N\lambda _1)\otimes V(N\lambda _2)$ 包含某个正整数 $N$ 的 $V(N\mu )$ 则 $V(d\lambda _1)\otimes V(d \lambda _2)$ 包含 $V(d\mu )$，假设 $\mu -\lambda _1-\lambda _2$ 属于根格。对于 $\mathfrak {g}={\mathfrak {sl}}_n$，著名的 Knutson-Tao 定理断言 $d=1$ 是一个饱和因子（参见 Knutson 和 Tao [J. Amer. Math. Soc. 12 （1999 年），第 1055-1090 页]）。更一般地，对于任何简单的李代数，明确的饱和因子是已知的。在 Kac-Moody 的情况下，$\Gamma _{\mathbb {N}}(\mathfrak {g})$ 不一定是有限生成的，因此这种因素的存在是先验的不清楚。在这里，我们获得了任何仿射 Kac-Moody Lie 代数的显式饱和因子。例如，在类型 $\tilde A_n$ 中，我们证明任何整数 $d\geq 2$ 都是饱和因子，概括 Brown 和 Kumar [Math. 安。360 (2014)，第 901-936 页]。

---