Face numbers of cubical barycentric subdivisions

TL;DR

This paper introduces the cubical barycentric subdivision for cubical complexes, provides formulas for associated h-vectors, and shows that key properties like symmetry and nonnegativity are preserved under subdivision.

Contribution

It defines the cubical barycentric subdivision and derives explicit formulas for the cubical h-vectors, demonstrating the preservation of important combinatorial properties.

Findings

Symmetry and nonnegativity of h-vectors are preserved.

Real rootedness of the short cubical h-polynomial is maintained.

Asymptotic behavior of h-vectors under successive subdivisions is characterized.

Abstract

The cubical barycentric subdivision sd_c(K) of a cubical complex K is introduced as an analogue of the barycentric subdivision of a simplicial complex. Explicit formulas for the short and long cubical h-vector of sd_c(K) are given, in terms of those of K. It is deduced that symmetry and nonnegativity of these h-vectors, as well as real rootedness of the short cubical h-polynomial, are preserved under cubical barycentric subdivision. The asymptotic behavior of the short and long cubical h-vectors of successive cubical barycentric subdivisions of K is also determined.