Perverse Equivalences and Dg-stable Combinatorics

TL;DR

This paper extends the theory of perverse equivalences to Calabi-Yau categories of negative dimension, linking it to mutation theory of simple-minded systems and providing a combinatorial model for Brauer tree algebras.

Contribution

It develops an analogue of perverse equivalences for negative Calabi-Yau categories and connects it to mutation theory, with explicit models for Brauer tree algebras.

Findings

Differential graded stable categories of symmetric algebras have negative Calabi-Yau dimension.

Perverse equivalences act transitively on bases in the dg-stable category of Brauer tree algebras.

The theory generalizes previous work from dimension -1 to arbitrary negative dimensions.

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.