Rationality of blocks of quasi-simple finite groups

TL;DR

This paper establishes upper bounds on Frobenius numbers of blocks of quasi-simple finite groups, showing their basic algebras are defined over small fields and confirming Donovan's conjecture for certain groups.

Contribution

It proves new bounds on Frobenius numbers of blocks and confirms Donovan's conjecture for blocks of special linear groups.

Findings

Morita Frobenius number of blocks ≤ 4

Strong Frobenius number ≤ 4|D|^2!

Donovan's conjecture holds for blocks of special linear groups

Abstract

Let $\ell$ be a prime number. We show that the Morita Frobenius number of an $\ell$-block of a quasi-simple finite group is at most 4 and that the strong Frobenius number is at most $4|D|^2!$, where D denotes a defect group of the block. We deduce that a basic algebra of any block of the group algebra of a quasi-simple finite group over an algebraically closed field of characteristic $\ell$ is defined over a field with $\ell^a$ elements for some $a \leq 4$. We derive consequences for Donovan's conjecture. In particular, we show that Donovan's conjecture holds for $\ell$-blocks of special linear groups.