Polytopal resolutions for finite groups

Abstract

For a finite group G acting faithfully on euclidean space we consider the convex hull of the orbit of a suitable vector. We show that the combinatorial structure of this polytope determines a polynomial growth free ℤG-resolution of ℤ. A resolution due to De Concini and Salvetti is recovered when G is a finite reflection group. A resolution based on the simplex is obtained from the regular representation of a finite group. □