Shellable Complexes from Multicomplexes

TL;DR

This paper characterizes the face numbers of Cohen-Macaulay subcomplexes of a specific join complex, providing necessary and sufficient conditions for their face numbers based on group actions.

Contribution

It proves that Novik's necessary conditions on face numbers are also sufficient, giving a complete characterization of face numbers for these complexes.

Findings

Necessary conditions are also sufficient for face numbers.

Complete characterization of face numbers of Cohen-Macaulay subcomplexes.

Extension of Novik's results to a full characterization.

Abstract

Suppose a group $G$ acts properly on a simplicial complex $\Gamma$. Let $l$ be the number of $G$-invariant vertices and $p_1, p_2, ... p_m$ be the sizes of the $G$-orbits having size greater than 1. Then $\Gamma$ must be a subcomplex of $\Lambda = \Delta^{l-1}* \partial \Delta^{p_1-1}*... * \partial \Delta^{p_m-1}$. A result of Novik gives necessary conditions on the face numbers of Cohen-Macaulay subcomplexes of $\Lambda$. We show that these conditions are also sufficient, and thus provide a complete characterization of the face numbers of these complexes.