Rationality of blocks of quasi-simple finite groups

Abstract

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