Quantum Cluster Variables via Serre Polynomials

TL;DR

This paper expresses quantum cluster variables using Serre polynomials of quiver Grassmannians, establishing counting polynomials and positivity for acyclic seeds, thus advancing the understanding of quantum cluster algebras.

Contribution

It introduces a new expression of quantum cluster variables via Serre polynomials, confirming conjectures and completing prior research in the field.

Findings

Existence of counting polynomials for quiver Grassmannians.

Positivity of quantum cluster monomials for acyclic seeds.

Confirmation of a recent conjecture by Rupel.

Abstract

For skew-symmetric acyclic quantum cluster algebras, we express the quantum $F$-polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.