Combined tilings and separated set-systems

• V. I. Danilov ^[1] ; A. V. Karzanov ^[2]
  1. [1] Central Institute of Economics and Mathematics of the RAS
  2. [2] Institute for System Analysis at FRC Computer Science and Control of the RAS
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 2, 2017, págs. 1175-1203
• Idioma: inglés
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• Resumen

  • 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.