Decomposable clutters and a generalization of Simon's conjectutre

TL;DR

This paper introduces decomposable clutters, a class of uniform clutters with ideals having linear quotients, and proves a conjecture related to shellability of simplicial complexes for certain dimensions.

Contribution

It defines decomposable clutters and proves a generalized version of Simon's conjecture for high-dimensional skeletons.

Findings

Decomposable clutters have ideals with linear quotients and resolutions.

Chordality of these clutters supports Simon's conjecture.

The conjecture is proven for dimensions d ≥ n-3.

Abstract

Each (equigenerated) squarefree monomial ideal in the polynomial ring $S=\mathbb{K}[x_1, \ldots, x_n]$ represents a family of subsets of $[n]$, called a (uniform) clutter. In this paper, we introduce a class of uniform clutters, called decomposable clutters, whose associated ideal has linear quotients and hence linear resolution over all fields. We show that chordality of these clutters guarantees the correctness of a conjecture raised by R. S. Simon on extendable shellability of $d$-skeletons of a simplex $\langle [n] \rangle$, for all $d$. We then prove this conjecture for $d \geq n-3$.