Positivity in Quantum Cluster Algebras and Flags of Valued Quiver Representations

TL;DR

This paper proves the positivity conjecture for adapted quantum cluster variables, provides explicit formulas for cluster monomials, and develops a method to decompose Grassmannians of subrepresentations in valued quiver representations.

Contribution

It offers a direct proof of positivity, explicit formulas for cluster monomials, and a general decomposition method for Grassmannians in valued quiver representations.

Findings

Positivity conjecture for adapted quantum cluster variables is proven.

Explicit formulas for all adapted cluster monomials are derived.

Decomposition of Grassmannians into products of standard vector space Grassmannians is achieved.

Abstract

In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of adapted cluster monomials. In particular, we describe all cluster monomials in cluster algebras and quantum cluster algebras of rank 2. One may obtain similar formulas for all finite type cluster monomials. The above results are achieved by computing explicit set-theoretic decompositions of Grassmannians of subrepresentations in adapted valued quiver representations into a disjoint union of products of standard vector space Grassmannians. We actually prove a more general result which should be of independent interest: we compute these decompositions for arbitrary flags of subrepresentations in adapted valued quiver representations. This implies the existence of counting polynomials for the number of points in these sets over different finite fields. Using this we extend the results of \cite{rupel} to quantum cluster algebras $\Acal_q(Q,\bfd)$, where $q$ is an indeterminate.