Coxeter Group - Homology

Homology

Since a Coxeter group W is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2-group, i.e. it is isomorphic to the direct sum of several copies of the cyclic group Z₂. This may be restated in terms of the first homology group of W.

The Schur multiplier M(W) (related to the second homology) was computed in (Ihara & Yokonuma 1965) for finite reflection groups and in (Yokonuma 1965) for affine reflection groups, with a more unified account given in (Howlett 1988). In all cases, the Schur multiplier is also an elementary abelian 2-group. For each infinite family {W_n} of finite or affine Weyl groups, the rank of M(W) stabilizes as n goes to infinity.