Geometric algebra for combinatorial geometries

AbstractGiven a combinatorial geometry (or “matroid”) M, defined on a finite set E, a certain abelian group T = TM, associated with M in a canonical fashion, is investigated. It is shown that T can be defined in terms of generators and relations in many possible ways, reflecting in an algebraic form the possibility of defining matroids in terms of either bases or hyperplanes or circuits or … and the combinatorial relations between these items. In addition, it is explained how T controls the representability of M as well as properties like regularity, binarity, ternarity, orientability etc. and how it relates to the universal representation ring of M. Further applications of this concept, relating it to (the highly non-trival part of) Tutte's homotopy theory for matroids and to matroids with coefficients are indicated and will be discussed in subsequent papers