Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm

Hernandez, David; Oya, Hironori

doi:10.1016/j.aim.2019.02.024

Cited by 23 publications

(33 citation statements)

References 52 publications

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“…These relations are t-deformations of the T -systems relations, first stated in [29]. These relations have not been generalized to nonsimply-laced types, except for type B n in [26]. This is the main reason why the results of this paper are limited to AD E types.…”

Section: Introductionmentioning

confidence: 98%

A quantum cluster algebra approach to representations of simply laced quantum affine algebras

Bittmann

2020

Math. Z.

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We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q, t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q, t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defined in a previous work as a quantum cluster algebra, contains indeed the well-known quantum Grothendieck ring of the category of finite-dimensional representations. Finally, we display our algorithm on a concrete example.

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Section: Introductionmentioning

confidence: 98%

A quantum cluster algebra approach to representations of simply laced quantum affine algebras

Bittmann

2020

Math. Z.

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“…However, for each

i \in I_0

the corresponding fundamental modules

V(\varpi _i)

are isomorphic to each other, since the corresponding fundamental weights are conjugate to each other under the Weyl group action (see [37, Section 5.2]). (b)Note that the Dynkin diagrams of type

B^{(1)}_2

and

A^{(2)}_3

in Table 1 are denoted by

C^{(1)}_2

and

D^{(2)}_3

in [27, pp. 54, 55], respectively. (c)Our conventions on quantum affine algebras are different from [23, 24]. To compare, we refer to [24, Remark 3.28].…”

Section: Quantum Groups Quantum Coordinate Rings and Quantum Affine A...mentioning

confidence: 99%

“…54, 55], respectively. (c)Our conventions on quantum affine algebras are different from [23, 24]. To compare, we refer to [24, Remark 3.28].…”

Section: Quantum Groups Quantum Coordinate Rings and Quantum Affine A...mentioning

confidence: 99%

Cluster algebra structures on module categories over quantum affine algebras

Kashiwara

Park

2022

Proceedings of London Math Soc

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We study monoidal categorifications of certain monoidal subcategories scriptCJ$\mathcal {C}_J$ of finite‐dimensional modules over quantum affine algebras, whose cluster algebra structures on their Grothendieck rings Kfalse(CJfalse)$K(\mathcal {C}_J)$ are closely related to the category of finite‐dimensional modules over quiver Hecke algebra of type A∞$A_\infty$ via the generalized quantum Schur–Weyl duality functors. In particular, when the quantum affine algebra is of type A$A$ or B$B$, the subcategory coincides with the monoidal category scriptCfrakturg0$\mathcal {C}_{\mathfrak {g}}^0$ introduced by Hernandez–Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.

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“…To conclude, we have to check that the powers of t match : this can be done using positivity in the quantum Grothendieck ring as in [HL2,Section 5.10] or directly as in [HO,section 9].…”

Section: 2mentioning

confidence: 99%