Brou\'e's abelian defect group conjecture holds for the Harada-Norton sporadic simple group $HN$

TL;DR

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

Contribution

It demonstrates that Broué's conjecture holds for all blocks of the Harada-Norton simple group, including a non-principal 3-block with an elementary abelian defect group.

Findings

Broué's conjecture verified for all primes for HN

Confirmed derived equivalence for specific 3-block

Extended validity of Broué's conjecture to HN

Abstract

In 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 $B$ of the normaliser $N_G(P)$ of $P$ in $G$ are derived equivalent (Rickard equivalent). This conjecture is called Brou\'e's abelian defect group conjecture. We prove in this paper that Brou\'e'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 Brou\'e's abelian defect group conjecture holds for all primes $p$ and for all $p$-blocks of the Harada-Norton simple group $HN$.