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Strongly Periodic Modules and Perverse Autoequivalences

Alfred Dabson

TL;DR

The paper develops a precise link between periodicity and perversity for finite-dimensional symmetric algebras by introducing strongly periodic modules and proving an if-and-only-if correspondence with two-step self-perverse autoequivalences of width $n$. It constructs a generalised periodic twist $\Phi_P$ from twisted strong periodicity data, showing it yields a two-step perverse equivalence and participates in a cycle of perverse autoequivalences. The results unify Grant’s periodicity framework with perverse equivalences in the derived category, and provide concrete realizations in blocks of the symmetric groups $\mathfrak{S}_6$ and $\mathfrak{S}_8$ in characteristic $3$. The work connects these representation-theoretic phenomena to Hochschild (co)homology and hints at a natural differential graded viewpoint, with potential extensions to broader settings.

Abstract

We introduce a notion of strong periodicity of a module over a finite-dimensional algebra over a field. We prove that the existence of such modules over certain idempotent algebras is both a necessary and sufficient condition for the existence of a two-step self-perverse equivalence of a finite-dimensional algebra. We survey some applications to the setting of the symmetric groups.

Strongly Periodic Modules and Perverse Autoequivalences

TL;DR

The paper develops a precise link between periodicity and perversity for finite-dimensional symmetric algebras by introducing strongly periodic modules and proving an if-and-only-if correspondence with two-step self-perverse autoequivalences of width . It constructs a generalised periodic twist from twisted strong periodicity data, showing it yields a two-step perverse equivalence and participates in a cycle of perverse autoequivalences. The results unify Grant’s periodicity framework with perverse equivalences in the derived category, and provide concrete realizations in blocks of the symmetric groups and in characteristic . The work connects these representation-theoretic phenomena to Hochschild (co)homology and hints at a natural differential graded viewpoint, with potential extensions to broader settings.

Abstract

We introduce a notion of strong periodicity of a module over a finite-dimensional algebra over a field. We prove that the existence of such modules over certain idempotent algebras is both a necessary and sufficient condition for the existence of a two-step self-perverse equivalence of a finite-dimensional algebra. We survey some applications to the setting of the symmetric groups.
Paper Structure (21 sections, 32 theorems, 48 equations)

This paper contains 21 sections, 32 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Symmetric algebras
  4. Derived equivalences
  5. Perverse equivalences
  6. Simple modules and projective modules
  7. Two-step perverse equivalences
  8. Standard equivalences
  9. Periodic Modules and Algebras
  10. Periodic modules
  11. Periodic and relatively periodic algebras
  12. A theorem of Grant
  13. Strongly Periodic Modules and Perverse Equivalences
  14. Strong periodicity
  15. The main theorem
  16. ...and 6 more sections

Key Result

Theorem 1

If $A$ is a finite-dimensional, symmetric $k$-algebra and \begin{tikzcd}[cramped,sep=small] \Phi:D^b(A) \arrow[r,"\sim"] & D^b(A) \end{tikzcd}is a two-step perverse equivalence of width $n$ satisfying a natural restriction condition, then there are projective $A$-modules $P$ and $Q$ such that,

Theorems & Definitions (55)

  • Theorem
  • Theorem
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Proposition 2.8
  • ...and 45 more