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 (J Am Math Soc 31(2):497–608, 2018). Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson–Thomas theory of quivers.
Highlights
Cluster algebras, a topic of significant, wide-ranging interest, were originally defined by Fomin and Zelevinsky [22] with the goal of better understanding Lusztig’s dual canonical bases [53] and total positivity [54]
There is a fan C forming a sub cone-complex of DAq, called the cluster complex, whose elements naturally correspond to the seeds Sj
As in [26], we deduce our results regarding cluster algebras from positivity results for scattering diagrams, which in turn are derived from positivity results in DT theory
Summary
A topic of significant, wide-ranging interest, were originally defined by Fomin and Zelevinsky [22] with the goal of better understanding Lusztig’s dual canonical bases [53] and total positivity [54]. The present paper uses ideas from Donaldson–Thomas (DT) theory [9,16,38,42,63,65] in concert with scattering diagram technology [14,21,25,26,32,34,43,45,57] to prove positivity in the quantum setting. We construct bases for various flavors of (skew-symmetric) quantum cluster A- and X -algebras, extending the set of quantum cluster variables, such that the structure constants for the multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the “quantum strong positivity conjecture.”. We use our constructions and results for quantum scattering diagrams to prove quantum analogues of all the main results of [26] for skew-symmetric quantum cluster algebras
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