The Mathieu group ${\sf M}_{12}$ vs ${\mathbf{SL_3(3)}}$ in characteristic 3

Shigeo Koshitani; Tetsuro Okuyama

The Mathieu group ${\sf M}_{12}$ vs ${\mathbf{SL_3(3)}}$ in characteristic 3

Representation Theory 2026-07-17 v1 Rings and Algebras

Authors: Shigeo Koshitani , Tetsuro Okuyama

Abstract

We prove that the principal $3$ -blocks of the Mathieu group ${\sf M_{12}}$ and the special linear group $\SL_3(3)$ are splendidly Rickard equivalent, and hence derived equivalent, by applying Rickard's theorem whose origin goes back to the second author. As a byproduct we answer to a question on the first Hochschild cohomology of the principal $3$ -blocks above posed by W. Murphy.

Keywords

Mathieu group special linear group principal block Rickard equivalence derived equivalence Hochschild cohomology modular representation

Cite

@article{arxiv.2607.15825,
  title  = {The Mathieu group ${\sf M}_{12}$ vs ${\mathbf{SL_3(3)}}$ in characteristic 3},
  author = {Shigeo Koshitani and Tetsuro Okuyama},
  journal= {arXiv preprint arXiv:2607.15825},
  year   = {2026}
}