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 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?
Keywords
Cite
@article{arxiv.1710.05314,
title = {On the Dowling and Rhodes lattices and wreath products},
author = {Stuart W. Margolis and John Rhodes and Pedro V. Silva},
journal= {arXiv preprint arXiv:1710.05314},
year = {2017}
}