Weight conjectures for Parker--Semeraro fusion systems

TL;DR

This paper proves that Parker--Semeraro fusion systems satisfy most of the Kessar--Linckelmann--Lynd--Semeraro weight conjectures, and confirms Robinson's weight conjecture for several specific groups and blocks.

Contribution

It establishes that Parker--Semeraro fusion systems meet six of nine key weight conjectures and verifies Robinson's conjecture for various principal blocks of finite groups.

Findings

Parker--Semeraro systems satisfy six of nine weight conjectures.

Robinson's weight conjecture holds for specific principal blocks of finite groups.

Results apply to groups like G_2(3), HN, BM, and others.

Abstract

We prove that the Parker--Semeraro systems satisfy six of the nine Kessar--Linckelmann--Lynd--Semeraro weight conjectures for saturated fusion systems. As a by-product we obtain that Robinson's ordinary weight conjecture holds for the principal $3$-block of Aut$(G_2(3))$, the principal $5$-blocks of $HN$, $BM$, Aut$(HN)$, $Ly$, the principal $7$-block of $M$, and the principal $p$-blocks of $G_2(p)$ for $p\geq 3 $.