On the Dowling and Rhodes lattices and wreath products

2017/10/15 by Margolis, Stuart W., Rhodes, John, Silva, Pedro V.
#05B35 #06B99 #18B40 #20E22 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1710.05314

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?

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