Brou\'e's abelian defect group conjecture holds for the double cover of the Higman-Sims sporadic simple group

TL;DR

This paper proves Broué's abelian defect group conjecture for the double cover of the Higman-Sims sporadic simple group, confirming the conjecture for all primes and blocks of this group.

Contribution

It establishes the validity of Broué's and Rickard's conjectures for the double cover of the Higman-Sims group, a significant case in sporadic simple groups.

Findings

Broué's conjecture holds for the double cover of Higman-Sims group.

Rickard's splendid equivalence conjecture is confirmed for this case.

The conjectures are verified for all primes and blocks of the group.

Abstract

In the representation theory of finite groups, there is a well-known and important conjecture, due to Brou\'e saying 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 B of the normaliser N_G(P) of P in G are derived equivalent. We prove in this paper, that Brou\'e's abelian defect group conjecture, and even Rickard's splendid equivalence conjecture are true for the faithful 3-block A with an elementary abelian defect group P of order 9 of the double cover 2.HS of the Higman-Sims sporadic simple group. It then turns out that both conjectures hold for all primes p and for all p-blocks of 2.HS.