Chromatic subdivision of simplicial complexes

The chromatic subdivision $\chi(\Delta)$ of a simplicial complex $\Delta$ is obtained by replacing the simplices of $\Delta$ by suitable Schlegel diagrams of the cross-polytope.

We investigate the $f$- and $h$-vector transformation moving from $\Delta$ to $\chi(\Delta)$ and its combinatorial interpretations in the same spirit as it was done for barycentric, edgewise and interval subdivisions.

Furthermore we study how algebraic invariants of the Stanley-Reisner ring as e.g, depth or projective dimension behave when passing from $\Delta$ to its chromatic subdivision $\chi(\Delta)$.