Rationality of blocks of quasi-simple finite groups
arXiv:1805.02015
The paper proves that for any ℓ‑block of a quasi‑simple finite group the Morita Frobenius number is at most 4 and the strong Frobenius number is bounded by 4|D|²!, leading to a description of basic algebras over small finite fields and confirming Donovan's conjecture for ℓ‑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.
Updated version with corrections of some typographical errors, addition of a result of Linckelmann (Proposition 2.6), and some clarifications in final proofs including changes to deal with very twisted groups (see Remark 4.6)