On the cd-index and gamma-vector of S*-shellable CW-spheres

TL;DR

This paper proves that the gamma-vector of the order complex of any polytope can be realized as the f-vector of a balanced simplicial complex, advancing understanding of the combinatorial structure of polytopes and CW-spheres.

Contribution

It establishes that the gamma-vector of the order complex of polytopes is the f-vector of a balanced simplicial complex, including all polytopes, via analysis of the cd-index and S-shellable spheres.

Findings

Gamma-vector equals f-vector of a balanced simplicial complex.

Certain parts of the cd-index are f-polynomials of colorable complexes.

Conjecture that the cd-index of a regular CW-sphere is a flag f-vector of a colored complex.

Abstract

We show that the $\gamma$-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing $c=1$ and considering the resulted polynomial in $d$, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.