# Hypersimplicial subdivisions

## TL;DR

This paper investigates the structure of subdivisions induced by linear projections of hypersimplices, revealing connections to secondary polytopes, answering a classical question positively in specific cases, and identifying non-lifting subdivisions in cyclic polytopes.

## Contribution

It characterizes fiber polytopes for hypersimplices, confirms a homotopy equivalence for polygons, and finds non-lifting subdivisions for cyclic polytopes, advancing understanding of hypersimplicial subdivisions.

## Key findings

- Fiber polytope is a Minkowski sum of secondary polytopes.
- Homotopy equivalence holds for polygons.
- Existence of non-lifting subdivisions in cyclic polytopes.

## Abstract

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