Vertex decomposability and weakly polymatroidal ideals

TL;DR

This paper explores the properties of simplicial complexes and monomial ideals, establishing equivalences between various algebraic and combinatorial conditions, and characterizing weakly polymatroidal ideals in terms of linear quotients and vertex splittability.

Contribution

It proves the equivalence of sequentially Cohen-Macaulay, shellable, and vertex decomposable conditions for matroidal ideals, and characterizes degree 2 monomial ideals as weakly polymatroidal if and only if they have linear quotients.

Findings

Equivalence of Cohen-Macaulay, shellability, and vertex decomposability for matroidal ideals.

Conditions under which simplicial complexes are vertex decomposable.

Characterization of degree 2 monomial ideals as weakly polymatroidal with linear quotients.

Abstract

Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\Delta$ be a simplicial complex on $n$ vertices and $I=I_{\Delta}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$ is a matroidal ideal then the following conditions are equivalent: $(i)$ $\Delta$ is sequentially Cohen-Macaulay; $(ii)$ $\Delta$ is shellable; $(iii)$ $\Delta$ is vertex decomposable. Also, if $I$ is a minimally generated by $u_1,\ldots,u_s$ such that $s\leq 3$ or ${\rm supp}(u_i)\cup {\rm supp}(u_j)=\{x_1,\ldots,x_n\}$ for all $i\neq j$, then $\Delta$ is vertex decomposable. Furthermore, we prove that if $I$ is a monomial ideal of degree $2$ then $I$ is weakly polymatroidal if and only if $I$ has linear quotients if and only if $I$ is vertex splittable.