Tensor products and q-characters of HL-modules and monoidal categorifications
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 581-619.

We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite-dimensional representations of a quantum affine algebra of type A. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type A with coefficients.

Dans ce travail, nous étudions certaines sous-catégories monoïdales (introduites par David Hernandez et Bernard Leclerc) de représentations de dimension finie d’une algèbre affine de type A. Nous classifions l’ensemble des représentations premières de ces sous-catégories, et donnons des conditions nécessaires et suffisantes pour que le produit tensoriel des deux représentations premières soit irréductible. Dans le cas où le produit tensoriel est réductible, nous décrivons une factorisation en modules premiers des facteurs simples. En conséquence, nous prouvons que ces sous-catégories monoïdales sont des catégorifications monoïdales d’algèbres amassées de type A avec coefficients.

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DOI: 10.5802/jep.101
Classification: 17B37,  20G42,  13F60
Keywords: Cluster algebra, monoidal categorification, prime representations
Brito, Matheus 1; Chari, Vyjayanthi 2

1 Departamento de Matemática, UFPR Curitiba - PR - Brazil, 81530-015
2 Department of Mathematics, University of California, Riverside 900 University Ave., Riverside, CA 92521, USA
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Brito, Matheus; Chari, Vyjayanthi. Tensor products and $q$-characters of HL-modules and monoidal categorifications. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 581-619. doi : 10.5802/jep.101. http://www.numdam.org/articles/10.5802/jep.101/

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