THE FACET IDEALS OF CHESSBOARD COMPLEXES

Abstract

We describe the irreducible decomposition of the facet ideal $\mathcal{ℱ}({\mathrm{\Delta }}_{m,n})$ of the chessboard complex ${\mathrm{\Delta }}_{m,n}$ with $n\ge m$. We also provide some lower bounds for the depth and regularity of the facet ideal $\mathcal{ℱ}({\mathrm{\Delta }}_{m,n})$. When $m\le 3$, we prove that these lower bounds can be obtained.