Strong positivity for quantum theta bases of quantum cluster algebras

Ben Davison, Travis Mandel

Research output: Contribution to journalArticlepeer-review

Abstract

We construct "quantum theta bases," extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross-Hacking-Keel-Kontsevich. Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson-Thomas theory of quivers.
Original languageEnglish
Number of pages85
JournalJournal of the London Mathematical Society
Early online date19 Jul 2021
DOIs
Publication statusE-pub ahead of print - 19 Jul 2021

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