Resumen de Combined tilings and separated set-systems

V. I. Danilov, A. Karzanov

• 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 ⊆ 2[n] (in particular, of the hypercube 2[n] itself, and the hyper-simplex {X ⊆ [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 combined (polygonal) tilings on a zonogon, which yields a new insight in the area.