Weight conjectures for fusion systems on an extraspecial group

TL;DR

The paper develops purely local counting conjectures for fusion systems and verifies them for saturated fusion systems on an extraspecial group $S$ of order $p^3$ and exponent $p$, including the Ruiz–Viruel exotic systems at $p=7$. It shows that nonconstrained fusion systems on $S$ admit only the trivial compatible family and computes local invariants $\\mathbf{m}$, $\\mathbf{w}$, and $\\mathbf{k}$ for the exotic systems, confirming seven conjectures that mirror local-to-global counting conjectures in block theory. Using these local results, the authors deduce Robinson's Ordinary Weight Conjecture for principal blocks of almost simple groups realizing such fusion systems and connect character-degree information of groups like $\\operatorname{PSL}_3(p)$ to these conjectures. The work thus links local fusion-system data to global block-theoretic phenomena and provides concrete verifications in the exotic $p$-fusion context.

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.

Theorems & Definitions (29)

• Conjecture 2.1
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• Lemma 4.1
• proof
• Lemma 4.2
• proof