The $f$- and $h$-vectors of Interval Subdivisions

TL;DR

This paper provides a detailed combinatorial analysis of how interval subdivisions affect the $f$- and $h$-vectors of simplicial complexes, proving the Charney-Davis conjecture under certain conditions.

Contribution

It offers a complete combinatorial description of the transformation matrices for $f$- and $h$-vectors under interval subdivision and establishes real-rootedness and conjecture validation.

Findings

Complete description of transformation matrices for $f$- and $h$-vectors.

Real-rootedness of the $h$-polynomial for complexes with non-negative $h$-vector.

Proof of the Charney-Davis conjecture for interval subdivisions with non-negative reciprocal $h$-vector.

Abstract

The interval subdivision Int$(\Delta)$ of a simplicial complex $\Delta$ was introduced by Walker. We give the complete combinatorial description of the entries of the transformation matrices from the $f$- and $h$-vectors of $\Delta$ to the $f$- and $h$-vectors of Int$(\Delta)$. We show that if $\Delta$ has non-negative $h$-vector then the $h$-polynomial of its interval subdivision has only real roots. As a consequence, we prove the Charney-Davis conjecture for Int$(\Delta)$, if $\Delta$ has non-negative reciprocal $h$-vector.