Shellability, Ehrhart Theory, and $r$-stable Hypersimplices

TL;DR

This paper introduces r-stable hypersimplices, studies their triangulations, and uses shellings and Ehrhart theory to connect these polytopes to graph polynomials and CR mappings of Lens spaces.

Contribution

It defines r-stable hypersimplices, shows they inherit triangulations from hypersimplices, and computes their Ehrhart polynomials, revealing new links to graph theory and geometry.

Findings

Triangulations of hypersimplices restrict to r-stable hypersimplices.

Shellings of these triangulations are constructed for the second hypersimplex.

Ehrhart h*-polynomials are computed using graph independence polynomials.

Abstract

Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n-set, we provide a shelling of this triangulation that sequentially shells each r-stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h*-polynomials of these polytopes, and the hypersimplex, via independence polynomials of graphs. For one such r-stable hypersimplex, this computation yields a connection to CR mappings of Lens spaces via Ehrhart-MacDonald reciprocity.