A Unified Treatment of the Geometric Algebra of Matroids and Even Δ-Matroids

Abstract

The concept of acombinatorial(W;P;U)-geometryfor a Coxeter groupW, a subsetPof its generating involutions and a subgroupUofWwithP⊆Uyields the combinatorial foundation for a unified treatment of the representation theories of matroids and of even Δ-matroids. The concept of a (W,P)-matroid as introduced by I. M. Gelfand and V. V. Serganova is slightly different, although for many important classes ofWandPone gets the same structures. In the present paper, we extend the concept of the Tutte group of an ordinary matroid to combinatorial (W;P;U)-geometries and suggest two equivalent definitions of a (W;P;U)-matroid with coefficients in a fuzzy ringK. While the first one is more appropriate for many theoretical considerations, the second one has already been used to show that (W;P;U)-matroids with coefficients encompass matroids with coefficients and Δ-matroids with coefficients. © 1999 Academic Press.