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Danilov, V. I.; Karzanov, A. V.; Koshevoy, G. A.

Combined tilings and separated set-systems. (English) Zbl 1359.05137

Sel. Math., New Ser. 23, No. 2, 1175-1203 (2017).
Summary: B. Leclerc and A. Zelevinsky [Transl., Ser. 2, Am. Math. Soc. 181(35), 85–108 (1998; Zbl 0894.14021)] 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 , so-called combined (polygonal) tilings on a zonogon, which yields a new insight in the area.

Cited in 12 Documents

MSC:

05E10 Combinatorial aspects of representation theory
05B45 Combinatorial aspects of tessellation and tiling problems
14M15 Grassmannians, Schubert varieties, flag manifolds
14G15 Finite ground fields in algebraic geometry

Keywords:

weakly separated sets; strongly separated sets; quasi-commuting quantum minors; rhombus tiling; Grassmann necklace

Citations:

Zbl 0894.14021
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Full Text: DOI arXiv

References:

[1] Danilov, V.I., Karzanov, A.V., Koshevoy, G.A.: Plücker environments, wiring and tiling diagrams, and weakly separated set-systems. Adv. Math. 224, 1-44 (2010) · Zbl 1219.05152 · doi:10.1016/j.aim.2009.10.017
[2] Danilov, V.I., Karzanov, A.V., Koshevoy, G.A.: On maximal weakly separated set-systems. J. Algebr. Comb. 32, 497-531. (2010). See also arXiv:0909.1423v1 [math.CO] (2009) · Zbl 1276.14071
[3] Danilov, V.I., Karzanov, A.V., Koshevoy, G.A.: The purity of separated set-systems related to Grassmann necklaces. arXiv:1312.3121 [math.CO] (2013) · Zbl 1393.05256
[4] Danilov, V.I., Karzanov, A.V., Koshevoy, G.A.: Combined tilings and the purity phenomenon on separated set-systems. arXiv:1401.6418 [math.CO] (2014) · Zbl 1359.05137
[5] Leclerc, B., Zelevinsky, A.: Quasicommuting families of quantum Plücker coordinates. Am. Math. Soc. Trans. Ser. 2 181, 85-108 (1998) · Zbl 0894.14021
[6] Oh, S., Postnikov, A., Speyer, D.E.: Weak separation and plabic graphs. arXiv:1109.4434 [math.CO] (2011) · Zbl 1309.05182
[7] Oh, S., Speyer, D.E.: Links in the complex of weakly separated collections. arXiv:1405.5191 [math.CO] (2014) · Zbl 1440.05256
[8] Postnikov, A.: Total positivity, Grassmannians, and networks. arXiv:math.CO/0609764 (2006)
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