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

TL;DR

This paper classifies all simple saturated fusion systems on specific $p$-groups derived from polynomial representations of $ ext{SL}_2(q)$, revealing new exotic systems and unifying known cases.

Contribution

It provides a complete classification of certain fusion systems related to polynomial representations of $ ext{SL}_2(q)$, including new exotic examples.

Findings

Classified all simple saturated fusion systems on the specified $p$-groups.

Included all known and new exotic fusion systems.

Unified existing fusion systems within a common framework.

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.