Generic Variables in Acyclic Cluster Algebras

TL;DR

This paper introduces generic variables in acyclic cluster algebras, explores their properties, and explicitly computes them for affine and Kronecker quivers, linking to known bases and bases comparison.

Contribution

It defines generic variables for acyclic cluster algebras, proves their relation to cluster monomials, and computes them explicitly for affine and Kronecker quivers.

Findings

Generic variables contain cluster monomials and coincide with them for Dynkin quivers.

Explicit formulas for generic variables in affine types.

The Kronecker quiver case yields a Z-basis comparable to known bases.

Abstract

Let $Q$ be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra $\mathcal A(Q)$. We prove that the set $\mathcal G(Q)$ of generic variables contains naturally the set $\mathcal M(Q)$ of cluster monomials in $\mathcal A(Q)$ and that these two sets coincide if and only if $Q$ is a Dynkin quiver. We establish multiplicative properties of these generic variables analogous to multiplicative properties of Lusztig's dual semicanonical basis. This allows to compute explicitly the generic variables when $Q$ is a quiver of affine type. When $Q$ is the Kronecker quiver, the set $\mathcal G(Q)$ is a $\mathbb Z$-basis of $\mathcal A(Q)$ and this basis is compared to Sherman-Zelevinsky and Caldero-Zelevinsky bases.