Paper detail

Cluster Algebras and Semi-invariant Rings II. Projections

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