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

David Hernandez, Hironori Oya

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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

Original languageEnglish
Pages (from-to)192-272
Number of pages81
JournalAdvances in Mathematics
Volume347
DOIs
Publication statusPublished - 2019 Apr 30

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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