Cluster Algebras and Semi-invariant Rings II. Projections

TL;DR

This paper explores how semi-invariant rings of quivers relate to cluster algebras when projecting through exceptional representations, establishing a connection between their algebraic structures and bases.

Contribution

It demonstrates that projections via exceptional representations preserve the upper cluster algebra structure and describes how their bases relate through quiver with potential models.

Findings

Semi-invariant rings are preserved under projections as upper cluster algebras.

The associated ice quivers are related through simple operations.

The basis of semi-invariant rings can be studied via quiver with potential models.

Abstract

Let ${\rm SI}_\beta(Q)$ be the semi-invariant ring of $\beta$-dimensional representations of a quiver $Q$. Suppose that $(Q,\beta)$ projects to another quiver with dimension vector $(Q',\beta')$ through an exceptional representation $E$. We show that if ${\rm SI}_\beta(Q)$ is the upper cluster algebra associated to an ice quiver $\Delta$, then ${\rm SI}_{\beta'}(Q')$ is the upper cluster algebra associated to $\Delta'$, where $\Delta'$ is obtained from $\Delta$ through simple operations depending on $E$. We also study the relation of their basis using the quiver with potential model.