Classification of Affine Symmetry Groups of Orbit Polytopes

TL;DR

This paper classifies the affine symmetry groups of orbit polytopes generated by finite group actions, revealing that generic orbit symmetry groups are consistent and determined by the group's character in characteristic zero, solving a longstanding problem.

Contribution

It provides a classification of affine symmetry groups of orbit polytopes and establishes the generic symmetry group structure in relation to the group's character in characteristic zero.

Findings

Generic symmetry groups are isomorphic for generic points.

In characteristic zero, the symmetry group is determined by the group's character.

The theory classifies all affine symmetry groups of vertex-transitive polytopes.

Abstract

Let $G$ be a finite group acting linearly on a vector space $V$. We consider the linear symmetry groups $\operatorname{GL}(Gv)$ of orbits $Gv\subseteq V$, where the \emph{linear symmetry group} $\operatorname{GL}(S)$ of a subset $S\subseteq V$ is defined as the set of all linear maps of the linear span of $S$ which permute $S$. We assume that $V$ is the linear span of at least one orbit $Gv$. We define a set of \emph{generic points} in $V$, which is Zariski-open in $V$, and show that the groups $\operatorname{GL}(Gv)$ for $v$ generic are all isomorphic, and isomorphic to a subgroup of every symmetry group $\operatorname{GL}(Gw)$ such that $V$ is the linear span of $Gw$. If the underlying characteristic is zero, "isomorphic" can be replaced by "conjugate in $\operatorname{GL}(V)$". Moreover, in the characteristic zero case, we show how the character of $G$ on $V$ determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (1977).