Graded decompositions of fusion products in rank 2

Barth, Leon; Kus, Deniz

doi:10.1215/21562261-2022-0016

Cited by 3 publications

(4 citation statements)

References 24 publications

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“…It is a very important and seemingly very hard problem to understand the fusion products of g-stable Demazure modules of various levels. One would like to find the generators and relations and the graded character of these modules, but very limited cases are known 3,41,53,90 .…”

Section: Review Articlementioning

confidence: 99%

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Quantum Affine Algebras, Graded Limits and Flags

Brito

Chari

Kus³

et al. 2022

J Indian Inst Sci

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In this survey, we review some of the recent connections between the representation theory of (untwisted) quantum affine algebras and the representation theory of current algebras. We mainly focus on the finite-dimensional representations of these algebras. This connection arises via the notion of the graded and classical limit of finite-dimensional representations of quantum affine algebras. We explain how this study has led to interesting connections with Macdonald polynomials and discuss a BGG-type reciprocity result. We also discuss the role of Demazure modules in this theory and several recent results on the presentation, structure and combinatorics of Demazure modules.

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Section: Review Articlementioning

confidence: 99%

“…However, we still have limited information on the structure of finite-dimensional Matheus Brito 1 , Vyjayanthi Chari 2 , Deniz Kus 3 and R. Venkatesh 4*…”

Section: Introductionmentioning

confidence: 99%

Quantum Affine Algebras, Graded Limits and Flags

Brito

Chari

Kus³

et al. 2022

J Indian Inst Sci

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“…Subsequently, by defining special quotients of the local Weyl modules, the conjecture was proved in certain special cases in [8,9,11,6,12,1]. In [6], a new family of finite-dimensional quotients of the local Weyl module called the Chari-Venkatesh (CV) modules was introduced.…”

Section: Introductionmentioning

confidence: 99%

“…1), we see that if (a, b) ∈ S ninv (λ + (j + 1)ω 2 , µ − (j + 1)ω 2 ), then (a + 1, b + 1)∈ S ninv (λ + jω 2 , µ − jω 2 ) unless b − a = λ 2 + j − µ 1 or b − a = µ 2 − λ 1 − j. Setting, ν (aj ,bj ) = λ + jω 2 − (µ 2 − j + a j − 2b j )ω 1 − (µ 1 + b j − 2a j )ω 2 , P 1 ninv (λ, µ) = {ν (a0,b0) : (a 0 , b 0 ) ∈ S ninv (λ, µ)} µ2−1 j=1 {ν (aj ,bj ) : (a j , b j ) ∈ S ninv (λ + jω 2 , µ − jω 2 ), b j − a j ∈ {λ 2 − µ 1 + j, µ 2 − λ 1 − j}}, .…”

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Fusion Product Modules for current algebra of type $A_2$

Khandai¹,

Rani²

2023

Preprint

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Fusion products of finite-dimensional cyclic modules that were defined in [7], form an important class of graded representations of current Lie algebras. In [6], a family of finite-dimensional indecomposable graded representations of the current Lie algebra called the Chari-Venkatesh(CV) modules, were introduced via generators and relations and it was shown that these modules are related to fusion products. In this paper, we study a class of these modules for current Lie algebras of type A 2 . By constructing a series of short exact sequences, we obtain a graded decomposition for them and show that they are isomorphic to fusion products of two finite-dimensional irreducible modules for current Lie algebras of sl 3 . Further, using the graded character of these CV-modules, we obtain an algebraic characterization of the Littlewood-Richardson coefficients that appear in the decomposition of tensor products of irreducible sl 3 (C)-modules.

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Prime representations in the Hernandez–Leclerc category: classical decompositions

Barth,

Kus

2023

Can. J. Math.-J. Can. Math.

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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.

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Graded decompositions of fusion products in rank 2

Cited by 3 publications

References 24 publications

Quantum Affine Algebras, Graded Limits and Flags

Quantum Affine Algebras, Graded Limits and Flags

Fusion Product Modules for current algebra of type $A_2$

Prime representations in the Hernandez–Leclerc category: classical decompositions

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