Rationality of blocks of quasi-simple finite 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 $a\le 4$. 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)