Splitting Polytopes

A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change

From weakly separated collections to matroid subdivisions

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 the case $(k,n) = (3,7)$ the set of arrangements of $(k-1)(n-k-1)$ weakly separated vertices of $\Delta_{k,n}$.Comment: 29 pages, 10 figures. v3: added proof of Corollary 3

Excluded minors for the class of split matroids

The class of split matroids arises by putting conditions on the system of split hyperplanes of the matroid base polytope. It can alternatively be defined in terms of structural properties of the matroid. We use this structural description to give an excluded minor characterisation of the class