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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the tensor semigroup of affine Kac-Moody lie algebras
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by Nicolas Ressayre HTML | PDF
J. Amer. Math. Soc. 35 (2022), 309-360 Request permission


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].

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Additional Information
  • Nicolas Ressayre
  • Affiliation: Institut Camille Jordan (ICJ), UMR CNRS 5208, Université Claude Bernard Lyon I, 43 boulevard du 11 novembre 1918, F-69622 Villeurbanne cedex, France
  • MR Author ID: 642966
  • Email:
  • Received by editor(s): August 31, 2017
  • Received by editor(s) in revised form: June 22, 2019, and October 1, 2020
  • Published electronically: September 9, 2021
  • Additional Notes: The author was partially supported by the French National Agency (Project GeoLie ANR-15-CE40-0012) and the Institut Universitaire de France (IUF)
  • © Copyright 2021 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 35 (2022), 309-360
  • MSC (2020): Primary 20G44, 14M15
  • DOI:
  • MathSciNet review: 4374953