Ragnar Freij-Hollanti: Combinatorial derived matroids

Let M be an arbitrary matroid with circuit set C(M). Gian-Carlo Rota and Henry Crapo asked, independently and with varous precise formulation, for a natural definition of a matroid delta(M) that has C(M) as its ground set. We propose, to our knowledge, the first definition of such a derived matroid that applies to arbitrary matroids, and is completely combinatorial. We generalize several of the known properties of a similar construction for represented matroids, given by Oxley and Wang. In particular, we show a bound on the rank, a decomposition into connected components, and a characterization of matroids for which delta(M)=M. We compute a family of examples, and formulate conjectures relating our construction to coajdjoints of geometric lattices.