Maximal Unimodular Systems of Vectors

A subset R of a vector space V(orRn) is called unimodular(or U -system) if every vector r ∈ R has an integral representation in every basis B ⊆ R. A U -system R is calledmaximal if one cannot add a non-zero vector not colinear to vectors of R such that the new system is unimodular and spans RR. In this work, we refine assertions of Seymour7and give a description of maximal U -systems. We show that a maximal U -system can be obtained as amalgams (as 1- and 2-sums) of simplest maximal U -systems called components. A component is a maximal U -system having no 1- and 2-decompositions. It is shown that there are three types of components: the root systems An, which are graphic, cographic systems related to non-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a special system E5representing the matroid R10from7which is neither graphic nor cographic. We give conditions that are necessary and sufficient for maximality of an amalgamated U -system. We give a complete description of all 11 maximal U -systems of dimension 6.