Coxeter-like complexes

Motivated by the Coxeter complex associated to a Coxeter system (W, S), we introduce a simplicial regular cell complex ∆(G, S) with a G-action associated to any pair (G, S) where G is a group and S is a finite set of generators for G which is minimal with respect to inclusion. We examine the topology of ∆(G, S), and in particular the representations of G on its homology groups. We look closely at the case of the symmetric group Sn minimally generated by (not necessarily adjacent) transpositions, and their type-selected subcomplexes. These include not only the Coxeter complexes of type A, but also the well-studied chessboard complexes.