Hypersimplicial subdivisions

Abstract

Let π:Rn→Rd be any linear projection, let A be the image of the standard basis. Motivated by Postnikov’s study of postitive Grassmannians via plabic graphs and Galashin’s connection of plabic graphs to slices of zonotopal tilings of 3-dimensional cyclic zonotopes, we study the poset of subdivisions induced by the restriction of π to the k-th hypersimplex, for k=1,…,n−1 . We show that: For arbitrary A and for k≤d+1 , the corresponding fiber polytope F(k)(A) is normally isomorphic to the Minkowski sum of the secondary polytopes of all subsets of A of size max{d+2,n−k+1} . When A=Pn is the vertex set of an n-gon, we answer the Baues question in the positive: the inclusion of the poset of π -coherent subdivisions into the poset of all π -induced subdivisions is a homotopy equivalence. When A=C(d,n) is the vertex set of a cyclic d-polytope with d odd and any n≥d+3, there are non-lifting (and even more so, non-separated) π -induced subdivisions for k=2.