Decomposable clutters and a generalization of Simon's conjectutre

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 $d \geq n-3$.