Hypersimplicial subdivisions

Abstract

AbstractLet $$\pi :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^d$$ π : R n → R d 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 $$\pi $$ π to the k-th hypersimplex, for $$k=1,\dots ,n-1$$ k = 1 , ⋯ , n - 1 . We show that: For arbitrary A and for $$k\le d+1$$ k ≤ d + 1 , the corresponding fiber polytope $${\mathcal {F}}^{(k)}(A)$$ F ( k ) ( A ) is normally isomorphic to the Minkowski sum of the secondary polytopes of all subsets of A of size $$\max \{d+2,n-k+1\}$$ max { d + 2 , n - k + 1 } . When $$A=\mathbf {P}_{n}$$ A = P n is the vertex set of an n-gon, we answer the Baues question in the positive: the inclusion of the poset of $$\pi $$ π -coherent subdivisions into the poset of all $$\pi $$ π -induced subdivisions is a homotopy equivalence. When $$A=\mathbf {C}(d,n)$$ A = C ( d , n ) is the vertex set of a cyclic d-polytope with d odd and any $$n \ge d+3$$ n ≥ d + 3 , there are non-lifting (and even more so, non-separated) $$\pi $$ π -induced subdivisions for $$k=2$$ k = 2 .