From weakly separated collections to matroid subdivisions

Abstract

We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex \(\Delta_{k,n}\), and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex \(e_J\) induces an \(\ell\)-split matroid subdivision of \(\Delta_{k,n}\), where \(\ell\) is the number of cyclic intervals in the \(k\)-element subset \(J\). We prove that a given collection of \(k\)-element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors, if and only if the arrangement of the blade \(((1,2,\ldots, n))\) on the corresponding vertices of \(\Delta_{k,n}\) induces a matroid (in fact, a positroid) subdivision. In this way we obtain a compatibility criterion for (planar) multi-splits of a hypersimplex, generalizing the rule known for 2-splits. We study in an extended example a matroidal arrangement of six blades on the vertices \(\Delta_{3,7}\). Mathematics Subject Classifications: 52B40, 05B45, 52B99, 05E99, 14T15Keywords: Combinatorial geometry, matroid subdivisions, weakly separated collections