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 ${\text {GL}}(Gv)$ of orbits $Gv\subseteq V$, where the linear symmetry group ${\text {GL}}(S)$ 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 generic points in V, which is Zariski open in V, and show that the groups ${\text {GL}}(Gv)$ for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group ${\text {GL}}(Gw)$ such that V is the linear span of Gw. If the underlying characteristic is zero, “isomorphic” can be replaced by “conjugate in ${\text {GL}}(V)$.” 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 (Geom Dedicata 6(3):331–337, 1977. https://doi.org/10.1007/BF02429904).