Brou\'e's isotypy conjecture for the sporadic groups and their covers and automorphism groups

TL;DR

This paper proves Broué's isotypy conjecture for blocks with abelian defect groups in certain sporadic groups and their covers, extending previous results from principal to non-principal blocks.

Contribution

It establishes Broué's conjecture for a new class of blocks in sporadic groups, covering non-principal blocks with specific group conditions.

Findings

Broué's isotypy conjecture holds for these sporadic group blocks.

Extends prior results from principal to non-principal blocks.

Provides new insights into block theory of sporadic groups.

Abstract

Let B be a p-block of a finite group G with abelian defect group D such that S\unlhd G, S'=S, G/Z(S)\le\Aut(S) and S/Z(S) is a sporadic simple group. We show that B is isotypic to its Brauer correspondent in N_G(D) in the sense of Brou\'e. This has been done by [Rouquier, 1994] for principal blocks and it remains to deal with the non-principal blocks.