On the Dowling and Rhodes lattices and wreath products

Plethysm for Wreath Products and Homology of Sub-Posets of Dowling Lattices

We prove analogues for sub-posets of the Dowling lattices of the results of Calderbank, Hanlon, and Robinson on homology of sub-posets of the partition lattices. The technical tool used is the wreath product analogue of the tensor species of Joyal.

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 l...

Quasi-Varieties, Congruences, and Generalized Dowling Lattices

Dowling lattices and their generalizations introduced by Hanlon are interpreted as lattices of congruences associated to certain quasi-varieties of sets with group actions. This interpretation leads, by a simple application of Mobius inversion, to polynomial identities which specialize to Hanlon's evaluation of the characteristic polynomials of generalized Dowling lattices. Analogous results ar...

Coupled Cells: Wreath Products and Direct Products

In this note we discuss the structure of systems of coupled cells (which we view as systems of ordinary differential equations) where symmetries of the system are obtained through the group G of global permutations of the cells and the group L of local internal symmetries of the dynamics in each cell. We show that even when the cells are assumed to be identical with identical coupling, the way ...

The Wreath Products