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Prime representations in the Hernandez–Leclerc category: classical decompositions

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  27 October 2023

Leon Barth  and
Deniz Kus
Leon Barth
Affiliation:
Faculty of Mathematics, Ruhr-University Bochum, Universitätsstraße 150, 44780 Bochum, Germany e-mail: leon.barth@rub.de
Deniz Kus*
Affiliation:
Faculty of Mathematics, Ruhr-University Bochum, Universitätsstraße 150, 44780 Bochum, Germany e-mail: leon.barth@rub.de
*
e-mail: deniz.kus@rub.de
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Abstract

We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez–Leclerc (HL) category for the quantum affine algebra associated with $\mathfrak {sl}_{n+1}$. When the HL category is realized as a monoidal categorification of a cluster algebra (Hernandez and Leclerc (2010, Duke Mathematical Journal 154, 265–341); Hernandez and Leclerc (2013, Symmetries, integrable systems and representations, 175–193)), these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with Brito, Chari, and Moura (2018, Journal of the Institute of Mathematics of Jussieu 17, 75–105), we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.

Keywords

Quantum affine algebrastruncated representationsconvex polytopes

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Secondary: 17B10: Representations, algebraic theory (weights)
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 32
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

D.K. was partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) (Grant No. 446246717).

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