On the Dowling and Rhodes lattices and wreath products

Abstract

Dowling and Rhodes defined different lattices on the set of triples (Subset, Partition, Cross Section) over a fixed finite group G. Although the Rhodes lattice is not a geometric lattice, it defines a matroid in the sense of the theory of boolean representable simplicial complexes. This turns out to be the direct sum of a uniform matroid with the lift matroid of the complete gain link graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar gain graph. This gives a new perspective on both matroids and also an application of matroid theory to the theory of finite semigroups. We also make progress on an important question for these classical matroids: what are the minimal boolean representations and the minimum degree of a boolean matrix representation?