Categorification and the quantum Grassmannian

Bernt Tore Jensen, Alastair King, Xiuping Su

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Abstract

In an earlier work, we gave an (additive) categorification of Grassmannian cluster algebras using the category CM(A) of Cohen-Macaulay modules for a certain Gorenstein order A. In this paper, for each cluster tilting object in CM(A), we construct a compatible pair (B,L) in a way that is consistent with mutation. This then determines a quantum cluster algebra and we show that, when (B,L) comes from a cluster tilting object with rank one summands, this quantum cluster algebra is (generically) isomorphic to the corresponding quantum Grassmannian. The key ingredient in the construction is a new invariant κ(M,N) of modules M,N in CM(A), which also has an intriguing link with mirror symmetry.
Original languageEnglish
Article number108577
JournalAdvances in Mathematics
Volume406
Early online date22 Jul 2022
DOIs
Publication statusPublished - 17 Sept 2022

Bibliographical note

Publisher Copyright:
© 2022 The Authors

Keywords

  • Categorification
  • Quantum Grassmannian
  • Quantum cluster algebra

ASJC Scopus subject areas

  • Mathematics(all)

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