On partial barycentric subdivision

TL;DR

This paper investigates the properties of the lth partial barycentric subdivision of simplicial complexes, extending previous work on standard barycentric subdivision, and analyzes how it affects combinatorial vectors like f- and h-vectors.

Contribution

It extends the understanding of barycentric subdivisions by analyzing the lth partial case and explores the associated transformation matrices for f- and h-vectors.

Findings

Derived properties of the transformation matrices for f- and h-vectors.

Extended previous results from standard to partial barycentric subdivisions.

Provided open problems for future research.

Abstract

The lth partial barycentric subdivision is defined for a (d-1)-dimensional simplicial complex \Delta and studied along with its combinatorial, geometric and algebraic aspects. We analyze the behavior of the f- and h-vector under the lth partial barycentric subdivision extending previous work of Brenti and Welker on the standard barycentric subdivision -- the case l = 1. We discuss and provide properties of the transformation matrices sending the f- and h-vector of \Delta to the f- and h-vector of its lth partial barycentric subdivision. We conclude with open problems.