Maurer's homotopy theory for even Δ-matroids and related combinatorial geometries

Abstract

I. M. Gelfand and V. V. Serganova introduced the concept of a (W, P)-matroid which is closely related to word problems in Coxeter groups and offers a unified approach to the theories of matroids, of Δ-matroids as well as to a large class of greedoids. In this paper, we study the slightly different concept of a combinatorial (W; P)-geometry, thereby unifying the exchange conditions in matroids and in even Δ-matroids. This approach is essentially built up graph theoretically and will have to serve as a combinatorial framework for a representation theory which will encompass the theory of matroids with coefficients as well as Δ-matroids representable by skew-symmetric matrices.