Weight conjectures for fusion systems on an extraspecial group

TL;DR

This paper verifies several local counting conjectures for fusion systems on an extraspecial group, including exotic systems at prime 7, and confirms Robinson's weight conjecture for related principal blocks in almost simple groups.

Contribution

It provides the first verification of these conjectures for fusion systems on an extraspecial group, including exotic cases, and connects them to block theory in finite groups.

Findings

Confirmed counting conjectures for fusion systems on extraspecial groups.

Verified Robinson's weight conjecture for principal blocks of almost simple groups.

Included validation for exotic fusion systems at prime 7.

Abstract

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here we verify some of these conjectures for fusion systems on an extraspecial group of order $p^3$, which contain among them the Ruiz-Viruel exotic fusion systems at the prime $7$. As a byproduct we verify Robinson's ordinary weight conjecture for principal $p$-blocks of almost simple groups $G$ realizing such (nonconstrained) fusion systems.