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

✍ Scribed by H.A. Jorge

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
222 KB
Volume
152
Category
Article
ISSN
0001-8708

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✦ Synopsis

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 P H 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(P H ), 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.

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