Partition and Cohen-Macaulay Extenders

TL;DR

This paper introduces a method to interpret the $h$-vector of non-partitionable complexes via partitionable extensions and explores conditions for Cohen-Macaulay extensions, connecting to shellability and Simon's conjecture.

Contribution

It constructs partitionable complexes extending non-partitionable ones and characterizes when Cohen-Macaulay extensions are possible, linking to shellability and matroid conjectures.

Findings

Any pure simplicial complex's $h$-vector can be expressed as a difference of two partitionable complexes' $h$-vectors.

Conditions are provided for when a Cohen-Macaulay extension of a complex exists.

The construction of Cohen-Macaulay extensions is straightforward under certain conditions.

Abstract

If a pure simplicial complex is partitionable, then its $h$-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex $\Delta$, we construct a complex $\Gamma \supseteq \Delta$ of the same dimension such that both $\Gamma$ and the relative complex $(\Gamma,\Delta)$ are partitionable. This allows us to rewrite the $h$-vector of any pure simplicial complex as the difference of two $h$-vectors of partitionable complexes, giving an analogous interpretation of the $h$-vector of a non-partitionable complex. By contrast, for a given complex $\Delta$ it is not always possible to find a complex $\Gamma$ such that both $\Gamma$ and $(\Gamma,\Delta)$ are Cohen-Macaulay. We characterize when this is possible, and we show that the construction of such a $\Gamma$ in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.