On the Dowling and Rhodes lattices and wreath products

TL;DR

This paper explores the lattice structures introduced by Dowling and Rhodes, revealing their connections to matroid theory and providing new insights into their representations and applications in finite semigroup theory.

Contribution

It establishes that the Rhodes lattice defines a matroid as a direct sum of a complete and a lift matroid, offering a new perspective and advancing understanding of Boolean representations.

Findings

Rhodes lattice forms a matroid as a direct sum of two matroids.

Dowling lattice defines the frame matroid over a biased graph.

Progress on minimal Boolean representations and their degrees.

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 complete matroid with a lift matroid of the complete biased graph over G. As is well known, the Dowling lattice defines the frame matroid over a similar biased 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?