Broué's abelian defect group conjecture holds for the Harada–Norton sporadic simple group HN

Abstract

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. 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 B of the normaliser NG(P) of P in G are derived equivalent (Rickard equivalent). This conjecture is called Broué's abelian defect group conjecture. We prove in this paper that Broué's abelian defect group conjecture is true for a non-principal 3-block A with an elementary abelian defect group P of order 9 of the Harada–Norton simple group HN. It then turns out that Broué's abelian defect group conjecture holds for all primes p and for all p-blocks of the Harada–Norton simple group HN.