2025/02/07 by Koshevoy, Gleb A., Li, Fang, Zhang, Lujun
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2502.05033
We adopt a formal and algebraic approach of Early \citeE2 to study the positive tropical Grassmannian Trop+ Grk,n. Specifically, we deal with positroid subdivision of hypersimplex induced by translated blades from any maximal weakly separated collection. One of our main results gives a necessary and sufficient condition on a maximal weakly separated collection to form a positroid subdivision of a hypersimplex corresponding to a simplicial cone in \rm Trop+Grk,n. For k = 2 our condition says that any weakly separated collection of two-elements sets gives such a simplicial cone, and all cones are of such a form. We also show that the maximality of any weakly separated collection is preserved under the boundary map, which armatively answers a question by Early in \citeE1. Plabic graphs, invented by Postnikov \citeP, are of use in proving this result. As a corollary, we get that all those positroid subdivisions are the finest. Thus, the flip of two maximal weakly separatedcollections corresponds to a pair of adjacent maximal cones in positive tropical Grassmannian.