Cubical subdivisions and local $h$-vectors

TL;DR

This paper extends the concept of local $h$-vectors from simplicial complexes to cubical complexes, defining cubical local $h$-vectors and exploring their properties and generalizations.

Contribution

It introduces cubical local $h$-vectors, establishing their properties and parallels with simplicial local $h$-vectors, and discusses their application to subdivisions of locally Eulerian posets.

Findings

Cubical local $h$-vectors are defined and shown to share properties with simplicial local $h$-vectors.

The theory applies to cubical subdivisions of cubes and complexes.

Generalizations to locally Eulerian posets are discussed.

Abstract

Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local $h$-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the $h$-vector of a simplicial complex is played by the (short or long) cubical $h$-vector of a cubical complex, defined by Adin, and the role of the local $h$-vector of a triangulation of a simplex is played by the (short or long) cubical local $h$-vector of a cubical subdivision of a cube. The cubical local $h$-vectors are defined in this paper and are shown to share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed.