An expansion formula for type A and Kronecker quantum cluster algebras

TL;DR

This paper develops an expansion formula for quantum cluster algebras of type A and Kronecker quivers, linking combinatorial structures to algebraic elements and revealing symmetry properties.

Contribution

It introduces a new expansion formula parametrized by perfect matchings and demonstrates a symmetry phenomenon in the Kronecker case.

Findings

Expansion formula parametrized by perfect matchings

Coefficients as q-powers with lattice-induced exponents

Reflectional symmetry satisfies Stembridge's q=-1 phenomenon

Abstract

We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker case, the coefficients are q-powers whose exponents are given by a weight function induced by the lattice of perfect matchings. As an application, we prove that a reflectional symmetry on the set of perfect matchings satisfies Stembridge's q=-1 phenomenon with respect to the weight function.