The Generalized Dowling Lattices

In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer n , a finite group G , and a meet sublattice K of the lattice of subgroups of G . For an appropriate choice of K the generalized Dowling lattice Dn{G,K) agrees with the ordinary Dowling lattice Dn(G). For a different choice of K, the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of G, generalizes the thick diagonal in CB . We compute the Möbius function and characteristic polynomial of the lattice Dn(G, K) along with the homology of Dn(G, K) in terms of the homology of K . We go on to compute the character of G-wrSn acting on the homology of Dn(G, K). This computation provides a nontrivial generalization of a result due to Stanley concerning the character of Sn acting on the top homology of the partition lattice.