Cohomology of Coxeter groups and Artin groups
For an irreducible Coxeter system \((W,S)\), with the group \(W\) finite, the authors construct an explicit free resolution \((C_*,\delta_*)\) of the trivial \(\mathbb{Z}[W]\)-module \(\mathbb{Z}\). In dimension \(k\), \(C_k\) is the free \(\mathbb{Z}[W]\)-module on the flags of subsets of \(S\) of cardinality \(k\). If \(n\) is the rank of \(W\), then there are \({n+k-1}\choose k\) such flags, so this resolution is much smaller than the standard resolution. Examples of calculations of cohomology using the resolution are included. The authors raise the question of when their resolution is the smallest possible, and this was answered by \textit{I. J. Leary} in his review of the article [in MR 2001f:20118].
scientific article; zbMATH DE number 1506008