Combined tilings and separated set-systems

Abstract

In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered n-element set [n] (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains ${\mathcal {D}}\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the hyper-simplex $\{X\subseteq [n]:|X|=m\}$ for m fixed), where ${\mathcal {D}}$ is called pure if all maximal weakly separated collections in ${\mathcal {D}}$ have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in $2^{[n]}$. This is obtained as a consequence of our study of a novel geometric–combinatorial model for weakly separated set-systems, so-called combined (polygonal) tilings on a zonogon, which yields a new insight in the area.