Quantum Laurent Positivity for Mutation-Acyclic Cluster Algebras

Abstract

We prove quantum Laurent positivity for mutation-acyclic skew-symmetrizable cluster algebras of arbitrary finite rank. For every integral compatible quantization with invertible frozen variables, each cluster variable has a Laurent expansion with coefficients in \(N[v^{\pm 1}]\) in every seed. For an indecomposable rigid representation of an acyclic valued quiver, we identify its Hall shuffle character with the character of a self-dual simple quiver Hecke module. These modules are real and admit affinizations; modules corresponding to the summands of a rigid direct sum strongly commute. A Feigin localization and a comparison of graded composition multiplicities establish positivity at an arbitrary target seed.

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