On the General Ranks of QP Representations

Abstract

We propose a mutation formula for the general rank from a principal component $\mathrm{PC}(\delta )$ of representations to another one $\mathrm{PC}(\epsilon )$ for a quiver with potential. We give sufficient conditions for the formula to hold. In particular, the formula holds when any of $\delta$ and $\epsilon$ is reachable. We discover several related mutation invariants.

arXiv comment30 pages, Comments are welcome. v2. typo fixing; v3. change most $\tau$ to $\hat{\tau }$; v4. no longer call it an "algorithm" + minor changes; final version to appear ART