Quantum grothendieck ring isomorphisms, cluster algebras and kazhdan-lusztig algorithm

David Hernandez, Hironori Oya

Research output: Contribution to journalArticlepeer-review

Abstract

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories CQ,Bn and CQ,A2n-1of finite-dimensional representations of quantum affine algebras of types B(1) n and A(1) 2n-1, respectively. 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 C2,Bnare given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

Original languageEnglish
JournalUnknown Journal
Publication statusPublished - 2018 Mar 18

Keywords

  • Dual canonical bases
  • Kazhdan-Lusztig algorithm
  • Quantum affine algebras
  • Quantum cluster algebras
  • Quantum Grothendieck rings
  • T-systems

ASJC Scopus subject areas

  • General

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