Chordality of Clutters with Vertex Decomposable Dual and Ascent of Clutters

TL;DR

This paper explores the properties of chordal clutters, especially those with vertex decomposable duals, and investigates conditions under which their associated ideals have linear quotients, contributing to the understanding of their algebraic and combinatorial structure.

Contribution

It introduces the concept of the ascent of a clutter and links chordality to vertex decomposability of the Alexander dual, advancing the theory of chordal clutters.

Findings

If the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, then the clutter is chordal.

The paper provides partial support for the conjecture that linear quotients imply chordality.

It splits a complex question into simpler parts using the ascent of a clutter.

Abstract

In this paper, we consider the generalization of chordal graphs to clutters proposed by Bigdeli, et al in J. Combin. Theory, Series A (2017). Assume that $\mathcal{C}$ is a $d$-dimensional uniform clutter. It is known that if $\mathcal{C}$ is chordal, then $I(\bar{\mathcal{C}})$ has a linear resolution over all fields. The converse has recently been rejected, but the following question which poses a weaker version of the converse is still open: "if $I(\bar{\mathcal{C}})$ has linear quotients, is $\mathcal{C}$ necessarily chordal?". Here, by introducing the concept of the ascent of a clutter, we split this question into two simpler questions and present some clues in support of an affirmative answer. In particular, we show that if $I(\bar{\mathcal{C}})$ is the Stanley-Reisner ideal of a simplicial complex with a vertex decomposable Alexander dual, then $\mathcal{C}$ is chordal.