The boolean complex of a Coxeter system, part 2

As Bridget discussed last week, the boolean complex of a Coxeter system is homotopy equivalent to a wedge spheres of maximal dimension. The number of spheres,called the boolean number, can be calculated recursively from the unlabeled Coxeter graph of the system. Specific computations reveal interesting enumerative properties of the boolean number, which appear closely linked with derangements. In this talk I will make this connection explicit. Given a (Coxeter) graph G we construct a set of derangements of the vertex set of G. Given such a derangement of the vertex set we construct a cycle in the homology of the boolean complex, which represent one of the spheres in a wedge sum. Combining these operations we obtain a basis for the homology indexed by the derangements, and thus we can give a combinatorial meaning to the spheres in the booolean complex in terms of derangements. Examples will be provided. This is joint work with Bridget Tenner.