Vertex decomposability and weakly polymatroidal ideals

Abstract

Let $K$ be a field and $R=K[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\hat{I}”$ be a simplicial complex on $n$ vertices and $I=I_{\hat{I}}”$ be its Stanley-Reisner ideal. In this paper, we show that if $I$ is a matroidal ideal then the following conditions are equivalent: $(i)$ $\hat{I}”$ is sequentially Cohen-Macaulay; $(ii)$ $\hat{I}”$ is shellable; $(iii)$ $\hat{I}”$ is vertex decomposable. Also, if $I$ is a minimally generated by $u_1,\dots ,u_s$ such that $s\le 3$ or $\mathrm{supp}(u_i)\cup \mathrm{supp}(u_j)=\{x_1,\dots ,x_n\}$ for all $i\ne j$, then $\hat{I}”$ 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.

8 pages, to appear in J. Algebraic Systems