Classification of affine symmetry groups of orbit polytopes

Abstract

A b s t r ac t . Let G be a finite group acting linearly on a vector space V . We consider the linear symmetry groups GL(Gv) of orbits Gv ⊆ V , where the linear symmetry group GL(S) of a subset S ⊆ 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 Zariskiopen in V , and show that the groups GL(Gv) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group GL(Gw) such that V is the linear span of Gw. If the underlying characteristic is zero, "isomorphic" can be replaced by "conjugate in 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).2010 Mathematics Subject Classification. Primary 52B15, Secondary 05E15, 20B25, 20C15.

Towards a Geometric Understanding of the 4-Dimensional Point Groups

“…One can try to visualize a point group G O(d) by looking at the orbit of some point 0 = v ∈ R d and taking the convex hull. This is called the G-orbit polytope of v. For an in-depth study of orbit polytopes and their symmetries, refer to [17,18].…”

“…A group is diploid if it contains the central reflection −id; otherwise, it is haploid. 18 In the classic classification, the diploid groups arise easily, but the haploid groups must be specially constructed as index-2 subgroups of diploid groups. Thus, the presence or absence of −id appears at the very beginning of the classic classification by quaternions.…”

Groups with a nontrivial nonideal kernel

“…(An orbit polytope is a polytope such that its (affine) symmetry groups acts transitively on the vertices of the polytope.) In joint work with Erik Friese [7] (continuing our earlier paper [6]), we develop a general theory, which shows, among other things, that G is isomorphic to the affine symmetry group of an orbit polytope when NKer R (G) = 1. When NKer R (G) > 1, this may or may not be the case.…”

Prescribing Symmetries and Automorphisms for Polytopes

“…The question of finding polytopes with prescribed automorphism group has also been asked as motivated by representation theory, see [Lad16,BL18,FL18]. These articles study orbit polytopes, that is, convex hulls of single point orbits under finite groups acting affinely on a real vector space.…”