Fusion systems related to polynomial representations of $\mathrm{SL}_2(q)$

TL;DR

The paper classifies core-free saturated fusion systems on polynomial $p$-groups $S_n(q)$ and $S_\Lambda(q)$ arising from polynomial representations of $\mathrm{SL}_2(q)$, showing that, for $1\le n\le p-1$ and $q>p$, every such fusion system is either a polynomial fusion system or an exotic one (including Henke–Shpectorov examples and a new infinite family). It extends the Clelland–Parker framework via polynomial modules $V_n(q)$ and $\Lambda(q)$, introduces new exotic families on $S_\Lambda(q)$, and develops a robust set of recognition and uniqueness results for the underlying groups and modules. The authors also construct and analyze pruned and extended polynomial fusion systems, establishing their saturation, exoticity, and closure properties. The results provide a near-complete landscape of core-free fusion systems on these $p$-groups and set the stage for further extensions to larger $p$-groups and discrete locality frameworks.

Abstract

Let $q$ be a power of a fixed prime $p$. We classify up to isomorphism all simple saturated fusion systems on a certain class of $p$-groups constructed from the polynomial representations of $\mathrm{SL}_2(q)$, which includes the Sylow $p$-subgroups of $\mathrm{GL}_3(q)$ and $\mathrm{Sp}_4(q)$ as special cases. The resulting list includes all Clelland--Parker fusion systems, a simple exotic fusion system discovered by Henke--Shpectorov, and a new infinite family of exotic examples.

Theorems & Definitions (155)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof