Brou\'e's abelian defect group conjecture holds for the sporadic simple Conway group Co_3

TL;DR

This paper proves Broué's abelian defect group conjecture for all blocks of the sporadic simple Conway group Co_3, confirming the conjecture's validity in this specific case.

Contribution

It verifies the strong version of Broué's conjecture for all primes and blocks of Co_3, a significant step in understanding modular representation theory of sporadic groups.

Findings

Proved the conjecture for the non-principal 2-block with defect group of order 8.

Confirmed the conjecture for all primes and blocks of Co_3.

Completes the verification of Broué's conjecture for Co_3.

Abstract

In the representation theory of finite groups, there is a well-known and important conjecture due to M. Brou\'e. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group P, then A and its Brauer corresponding block A_N of the normaliser N_G(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Strong Version of Brou\'e's Abelian Defect Group Conjecture. In this paper, we prove that the strong version of Brou\'e's abelian defect group conjecture is true for the non-principal 2-block A with an elementary abelian defect group P of order 8 of the sporadic simple Conway group Co_3. This result completes the verification of the strong version of Brou\'e's abelian defect group conjecture for all primes p and for all p-blocks of Co_3.