Cluster Algebras and Semi-invariant Rings II. Projections

Abstract

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

arXiv comment28 pages, 5 figures. text overlap with arXiv:1504.02970, arXiv:1411.4693. v2. fix typos, and add one conjecture

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