Cohomology of Coxeter groups

Let W be a finitely generated Coxeter group. We describe a method which is useful in computing the cohomology of W with local coefficients. It is based on the determination of an explicit combinatorial resolution of Z over Z[W] (developed in a previous paper) which grows polynomially with the dimension. Let S be a Coxeter set of generators for W: in dimension k, the rank of the resolution equals the number of flags Γ1⊃Γ2⊃⋯, Γi⊂S, having cardinality k, while the coboundary is explicitly given in terms of minimal coset representatives of W. e use such a method in order to produce some cohomologies in the cases of dihedral groups and symmetric groups.