Maximal unimodular systems of vectors (Q1306930)

Assertions of \textit{P. D. Seymour} [J. Comb. Theory, Ser. B 28, 305-359 (1980; Zbl 0443.05027)] on unimodular systems of vectors (or \(U\)-systems) are refined and applied to maximal \(U\)-systems. A maximal \(U\)-system can be obtained as amalgams of simplest components. Three types of components exist: the root systems \(A_n\), the cographic systems of non-planar 3-connected cubic graphs, and the systems \(E_5\) representing the matroid \(R_{10}\). Necessary and sufficient conditions are derived for maximality of an amalgamated \(U\)-system. All 11 maximal \(U\)-systems of dimension 6 are described.