Graded quiver varieties, quantum cluster algebras and dual canonical basis

TL;DR

This paper explores the interplay between graded quiver varieties, quantum cluster algebras, and dual canonical bases, providing new categorifications, verifying positivity conjectures, and extending previous results to acyclic cases.

Contribution

It introduces new categorifications of acyclic quantum cluster algebras, verifies the positivity conjecture in certain cases, and generalizes prior work to the acyclic setting.

Findings

Deformed monoidal categorifications of acyclic quantum cluster algebras.

Verification of the quantum positivity conjecture with acyclic seeds.

All quantum cluster monomials belong to the dual canonical basis in the studied setting.

Abstract

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal categorifications of acyclic quantum cluster algebras with specific coefficients. In particular, the (quantum) positivity conjecture is verified whenever there is an acyclic seed in the (quantum) cluster algebra. In the second part of the paper, we introduce new quantizations and show that all quantum cluster monomials in our setting belong to the dual canonical basis of the corresponding quantum unipotent subgroup. This result generalizes previous work by Lampe and by Hernandez-Leclerc from the Kronecker and Dynkin quiver case to the acyclic case. The Fourier transform part of this paper provides crucial input for the second author's paper where he constructs bases of acyclic quantum cluster algebras with arbitrary coefficients and quantization.