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Simplicial complexes of whisker type

Let $I\subset K[x_1,\ldots,x_n]$ be a zero-dimensional monomial ideal, and $Δ(I)$ be the simplicial complex whose Stanley--Reisner ideal is the polarization of $I$. It follows from a result of Soleyman Jahan that $Δ(I)$ is shellable. We give a new short proof of this fact by providing an explicit shelling. Moreover, we show that $Δ(I)$ is even vertex decomposable. The ideal $L(I)$, which is defined to be the Stanley--Reisner ideal of the Alexander dual of $Δ(I)$, has a linear resolution which is cellular and supported on a regular CW-complex. All powers of $L(I)$ have a linear resolution. We compute $\mathrm{depth}\ L(I)^k$ and show that $\mathrm{depth}\ L(I)^k=n$ for all $k\geq n$.