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Ellis, G,Harris, J,Skoldberg, E
Journal Fur Die Reine Und Angewandte Mathematik
Polytopal resolutions for finite groups
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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.Our aim in this paper is to explain how, for any finite group G, a finite calculation involving convex hulls leads to an explicit recursive description of all dimensions of a free ZG-resolution in which the number of generators grows polynomially with dimension.Let alpha : G -> GL(R-n) be a faithful representation of a finite group G. Let v is an element of R-n be a point such that alpha(g)v not equal v for all 1 not equal g is an element of G. Such a point is said to be in generalposition and exists since F(g) = { w is an element of R-n : alpha(g)w = w} is a vector space of dimension less than n for any 1 not equal g is an element of G. Take v to be any point contained in no subspace F(g) for 1 not equal g is an element of G. We define P(G) = P(G, alpha, v) to be the convex hull of the points in the orbit v(G) = {alpha(g)v: g is an element of G). The face lattice of P(G) depends in general on the choice of v as well as alpha. This lattice is described for a range of groups in a forthcoming paper [2].
DOI 10.1515/CRELLE.2006.071
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