Maximal Unimodular Systems of Vectors

✍ Scribed by Vladimir Danilov; Viatcheslav Grishukhin

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 249 KB
• Volume: 20
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1999.0298

✦ Synopsis

A subset R of a vector space V (or R n ) 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 called maximal 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 Seymour [7] and 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 A n , which are graphic, cographic systems related to non-planar 3-connected cubic graphs without separating cuts of cardinality 3, and a special system E 5 representing the matroid R 10 from [7] which 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.