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

Abstract

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms specialize at $t=1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

arXiv commentv1 : 63 pages. v2 : 64 pages. Minor modification. To appear in Adv. Math

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Cited by in corpus (8)

• Monoidal categorification and quantum affine algebras
• Q-data and representation theory of untwisted quantum affine algebras
• Isomorphisms among quantum Grothendieck rings and propagation of positivity
• Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras
• The $(q,t)$-Cartan matrix specialized at $q=1$
• Equivalence between module categories over quiver Hecke algebras and Hernandez-Leclerc's categories in general types
• Inflations among quantum Grothendieck rings of type A
• Quantum periodicity and Kirillov-Reshetikhin modules