Classification of Affine Symmetry Groups of Orbit 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).