Weight conjectures for Parker–Semeraro fusion systems

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 $\mathrm{Aut}\left(G_2\right(3\left)\right)$, the principal 5-blocks of HN, BM, $\mathrm{Aut}\left(HN\right)$, Ly, the principal 7-block of M, and the principal p-blocks of $G_2\left(p\right)$ for $p\ge 3$.