Beitr\"age zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 37 (1996), No. 1, 161-198.

A Unified Treatment of the Theories of Matroids with Coefficients and
of \bigdelta-Matroids with Coefficients

Walter Wenzel

Abstract.

The representation theory of matroids and that of even $\Delta $-matroids
have much in common. In this paper, a unified representation theory is
developed which will also encompass the theory of matroids with coefficients
and that of $\Delta $-matroids with coefficients. This is established by
identifying set systems with special cosets in Coxeter groups as was done by
I.\1 M.\1 Gelfand and V.\1 V.\1 Serganova in their work concerning
$(W,P)$-{\it matroids}\/; however, in order to extend the theory of matroids
with coefficients it was necessary to study a slightly different concept,
named a {\it Combinatorial $(W;P;U)$-geometry} which is defined for a Coxeter
group $W$, a subset $P$ of its generating involutions, and some subgroup
$U$ of $W$ containing~$P$.

Keywords and phrases:
Representation theory of matroids, matroids with coefficients,
Grassmann-Pl\"ucker relations, $\Delta $-matroids, skew-symmetric matrices,
Pfaffian forms, the base graph of a matroid, antipodal graphs, convexity in
graphs, Coxeter groups.

MSC 1991: 05B25, 05B35, 20B25, 51D20 (Primary)

          15A03, 15A15, 51D10, 51F15, 52A30, 68E10 (Secondary)