A balanced non-partitionable Cohen-Macaulay complex

TL;DR

This paper constructs balanced Cohen-Macaulay complexes that counter the longstanding conjecture by Stanley, demonstrating that not all such complexes are partitionable, thus providing new insights into their structural properties.

Contribution

The authors provide the first known balanced Cohen-Macaulay complexes that are non-partitionable, disproving Stanley's conjecture and answering a key open question.

Findings

Counterexamples are balanced Cohen-Macaulay complexes.

Stanley's conjecture on partitionability is false.

Balanced complexes can be non-partitionable.

Abstract

In a recent paper, Duval, Goeckner, Klivans and Martin disproved the longstanding conjecture by Stanley, that every Cohen-Macaulay simplicial complex is partitionable. We construct counterexamples to this conjecture that are even \emph{balanced}, i.e., their underlying graph has a minimal coloring. This answers a question by Duval et al. in the negative.