Orthogonal groups in characteristic 2 acting on polytopes of high rank

TL;DR

This paper demonstrates that orthogonal and symplectic groups over fields of characteristic 2 can act on high-rank abstract regular polytopes, revealing new symmetries in geometric structures.

Contribution

It establishes the existence of high-rank regular polytopes acted upon by orthogonal and symplectic groups in characteristic 2, expanding understanding of their geometric actions.

Findings

Orthogonal groups $ ext{Orth}^{ ext{±}}(2m, extbf{F}_k)$ act on rank $2m$ polytopes.

Symplectic groups $ ext{Sp}(2m, extbf{F}_k)$ act on rank $2m+1$ polytopes.

Results hold for all integers $m ext{,}k ext{ with } m,k ext{ ≥ 2}.

Abstract

We show that for all integers $m\geq 2$, and all integers $k\geq 2$, the orthogonal groups $\Orth^{\pm}(2m,\Fk)$ act on abstract regular polytopes of rank $2m$, and the symplectic groups $\Sp(2m,\Fk)$ act on abstract regular polytopes of rank $2m+1$.