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Regular version of the site

Article

Combined tilings and separated set-systems

Selecta Mathematica, New Series. 2017. Vol. 23. No. 2. P. 1175-1203.
Danilov V., Karzanov A., G.A.Koshevoy.

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]D⊆2[n] (in particular, of the hypercube 2[n]2[n] itself, and the hyper-simplex {X⊆[n]:|X|=m}{X⊆[n]:|X|=m} for m fixed), where DD is called pure if all maximal weakly separated collections in DD 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]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 (polygonaltilingson a zonogon, which yields a new insight in the area.