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 $D\subseteq 2^{[n]}$ (in particular, of the hypercube $2^{[n]}$ itself, and the hyper-simplex $\{X\subseteq [n]:|X|=m\}$ for $m$ fixed), where $D$ is called pure if all maximal weakly separated collections in $D$ 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 \emph{combined (polygonal) tilings} on a zonogon, which yields a new insight in the area.

30 pages. Revised version. To appear in Selecta Mathematica

Cited by in corpus (3)

• Plabic graphs and zonotopal tilings
• Cubillages of cyclic zonotopes
• Weak Separation, Pure Domains and Cluster Distance