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arXiv · 1911.12791

Partition and Cohen-Macaulay Extenders

Joseph Doolittle·Bennet Goeckner·Alexander Lazar
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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.

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Joseph Doolittle, Bennet Goeckner, Alexander Lazar. 2019-11-28. Partition and Cohen-Macaulay Extenders. https://arxiv.org/abs/1911.12791

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