2019/10/25 by Early, Nick
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.1910.11522
We study arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex Δk,n, and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex eJ induces an ℓ-split matroid subdivision of Δk,n, where ℓ 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,…, n)) on the corresponding vertices of Δk,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 Δk,n.