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? This is joint work with Stuart Margolis (Bar Ilan University, Israel) and John Rhodes (University of California at Berkeley, USA).
On the Dowling and Rhodes lattices and wreath products.
Room FC1 030, DMat-FCUP at 14:30
Friday, 19 January, 2018 (All day)