Combinatorics of Polytopes with a Group of Linear Symmetries of Prime Power Order

Abstract

Generalizing a result of Stanley [13] on centrally symmetric polytopes, Adin [2], [3] has derived tight lower bounds for the face numbers of a rational simplical polytope equipped with a fixed-point-free linear action of a cyclic group G of prime power order. The main goal of this paper is to extend these results further by replacing Adin's fixedpoint-free condition with the assumption that the action of G is proper. As corollaries, we obtain a generalization of Adin's equivariant lower bound theorem and of a condition by Stanley [13] implying combinatorial isomorphism with a minimal polytope. Finally, we prove sufficiency of an equivariant version of the McMullen and Walkup generalized lower bound conjecture.

A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes

“…Our starting point was Stanley's result [21] asserting that a cs simplicial d-polytope P satisfies g r (P ) ≥ d r − d r−1 for all 1 ≤ r ≤ ⌊d/2⌋. However, it is worth mentioning that Adin [1] and later Jorge [12] showed that Stanley's result can be suitably extended to polytopes with more intricate symmetries. It would be very interesting to check if our techniques can be adapted to provide a characterization of polytopes that minimize g 2 in these more general settings.…”

Representations of cyclic groups acting on complete simplicial fans

“…This suggests the possibility of arriving at properties of ∆ via investigation of the characters. In fact, work of this this flavor (using more directly the isomorphism of H * (X σ , Q) and Q[∆]/Θ in order to use additional properties of the toric variety) has been done to investigate simplicial polytopes with certain simple symmetries, see, e.g., [1], [4], and [5].…”