Combinatorics of antiprism triangulations

TL;DR

This paper explores the combinatorial and algebraic properties of antiprism triangulations of simplicial complexes, including their effects on $h$-vectors and Stanley--Reisner rings, revealing real-rootedness and Lefschetz properties.

Contribution

It provides new insights into the enumerative and algebraic invariants of antiprism triangulations, including their impact on $h$-vectors and Lefschetz properties.

Findings

The $h$-polynomial of antiprism triangulation of a simplex is real-rooted.

Antiprism triangulation of a shellable complex has the almost strong Lefschetz property over ${ m f R}$.

Enumerative invariants of antiprism triangulations are characterized and related to algebraic properties.

Abstract

The antiprism triangulation provides a natural way to subdivide a simplicial complex $\Delta$, similar to barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the simplicial complex of chains of multi-pointed faces of $\Delta$, from a combinatorial point of view, and by successively applying the antiprism construction, or balanced stellar subdivisions, on the faces of $\Delta$, from a geometric point of view. This paper studies enumerative invariants associated to this triangulation, such as the transformation of the $h$-vector of $\Delta$ under antiprism triangulation, and algebraic properties of its Stanley--Reisner ring. Among other results, it is shown that the $h$-polynomial of the antiprism triangulation of a simplex is real-rooted and that the antiprism triangulation of $\Delta$ has the almost strong Lefschetz property over ${\mathbb R}$ for every shellable complex $\Delta$. Several related open problems are discussed.