Smith-Type Inequalities for a Polytope with a Solvable Group of Symmetries

Abstract

The object of this paper is to obtain a set of inequalities relating the face numbers of different orbit types of a simplicial polytope P with a finite solvable group G of linear symmetries. It is assumed that (1) for each subgroup H of G, the fixed point set PH is a subpolytope of P, and (2) the toric variety X(P) associated to P is nonsingular. The action of G on P induces an action on X(P), and we describe a set of Smith-type inequalities between the Betti numbers of X(P)H, where H ranges through the set of subgroups of G. By relating each X(P)H with X(PH), we then express these inequalities in terms of the face numbers of the different orbit types of P and the rank of fixed point sets of certain compact tori. This rank is determined explicitly when G is abelian. Moreover, assumption (2) is removed for a polytope of dimension 2.