Geometric algebra for combinatorial geometries
The starting point of this paper is a correspondence in which with a given combinatorial geometry, or matroid, M defined on a finite set E a certain abelian group \(\Pi_ m\) is associated in a canonical fashion. This group is called the Tutte-group of M, and the study of its structure is the main purpose of the paper. Particularly the following two problems are discussed. First, there is a group of theorems describing how \(\Pi_ M\) can be defined in terms of generators and relations in many possible ways using either the basis of M or its hyperplanes or its circuits etc. Second, another group of theorems explains how \(\Pi_ M\) controls the representability of M as well as its properties like regularity, binarity, ternarity, orientability etc., and how it relates to the universal representation ring of M. Other more detailed results concerning the structure of Tutte-groups can be found in three forthcoming papers briefly characterized in the text, this paper is announced to be the first and introductory in the series of papers.
scientific article; zbMATH DE number 4120185
scientific article