Face vectors of subdivided simplicial complexes

TL;DR

This paper studies the behavior of face vectors of subdivided simplicial complexes, revealing convergence and symmetry properties of their roots, and extends these results to various subdivision methods using algebraic and geometric techniques.

Contribution

It provides a new proof, uncovers symmetry in the roots, and generalizes the results to arbitrary subdivision methods with explicit formulas for barycentric subdivision.

Findings

Roots of face vectors converge to fixed values depending on dimension.

Roots exhibit symmetry about -2.

Results apply to various subdivision methods with explicit formulas.

Abstract

Brenti and Welker have shown that for any simplicial complex X, the face vectors of successive barycentric subdivisions of X have roots which converge to fixed values depending only on the dimension of X. We improve and generalize this result here. We begin with an alternative proof based on geometric intuition. We then prove an interesting symmetry of these roots about the real number -2. This symmetry can be seen via a nice algebraic realization of barycentric subdivision as a simple map on formal power series in two variables. Finally, we use this algebraic machinery with some geometric motivation to generalize the combinatorial statements to arbitrary subdivision methods: any subdivision method will exhibit similar limit behavior and symmetry. Our techniques allow us to compute explicit formulas for the values of the limit roots in the case of barycentric subdivision.