From weakly separated collections to matroid subdivisions

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