Square-free powers of Cohen-Macaulay simplicial forests

Let $I(\mathrm{\Delta }{)}^{[k]}$ denote the $k^{\text{th}}$ square-free power of the facet ideal of a simplicial complex $\mathrm{\Delta }$ in a polynomial ring $R$. Square-free powers are intimately related to the Matching Theory' and Ordinary Powers’. In this article, we show that if $\mathrm{\Delta }$ is a Cohen-Macaulay simplicial forest, then $R\mathrm{/}I(\mathrm{\Delta }{)}^{[k]}$ is Cohen-Macaulay for all $k\ge 1$. This result is quite interesting since all ordinary powers of a graded radical ideal can never be Cohen-Macaulay unless it is a complete intersection. To prove the result, we introduce a new combinatorial notion called special leaf, and using this, we provide an explicit combinatorial formula of $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(R\mathrm{/}I(\mathrm{\Delta }{)}^{[k]})$ for all $k\ge 1$, where $\mathrm{\Delta }$ is a Cohen-Macaulay simplicial forest. As an application, we show that the normalized depth function of a Cohen-Macaulay simplicial forest is nonincreasing.