Cluster Algebras and Semi-invariant Rings II. Projections

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