Pfaffian forms and Δ-matroids

Abstract

In this paper it is shown that skew-symmetric n × n-matrices with coefficients in a field K correspond via Pfaffian forms in a canonical one-to-one fashion to K-valued maps defined on the power set B ({1,…, n}), which satisfy certain identities. As an application, we describe representability of Δ-matroids by skew-symmetric matrices in terms of these maps. This suggests a definition of orientable and valuated Δ-matroids or, more generally, of Δ-matroids with coefficients which is analogous to the corresponding concept studied in matroid theory.