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

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.