Perverse Equivalences and Dg-stable Combinatorics

Abstract

Chuang and Rouquier describe an action by perverse equivalences on the set of bases of a triangulated category of Calabi-Yau dimension $-1$. We develop an analogue of their theory for Calabi-Yau categories of dimension $w<0$ and show it is equivalent to the mutation theory of $w$-simple-minded systems. Given a non-positively graded, finite-dimensional symmetric algebra $A$, we show that the differential graded stable category of $A$ has negative Calabi-Yau dimension. When $A$ is a Brauer tree algebra, we construct a combinatorial model of the dg-stable category and show that perverse equivalences act transitively on the set of $|w|$-bases.