On maximal weakly separated set-systems
Vladimir I. Danilov
, Alexander V. Karzanov
and Gleb A. Koshevoy
DOI: 10.1007/s10801-010-0224-x
Abstract
For a permutation ω \in S n , Leclerc and Zelevinsky in Am. Math. Soc. Transl., Ser. 2 181, 85-108 ( 1998) introduced the concept of an ω - chamber weakly separated collection of subsets of {1,2,\cdots , n} and conjectured that all inclusionwise maximal collections of this sort have the same cardinality \ell ( ω )+ n+1, where \ell ( ω ) is the length of ω . We answer this conjecture affirmatively and present a generalization and additional results.
Pages: 497–531
Keywords: keywords weakly separated sets; rhombus tiling; generalized tiling; weak Bruhat order; cluster algebras
Full Text: PDF
References
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4. Danilov, V., Karzanov, A., Koshevoy, G.: Plücker environments, wiring and tiling diagrams, and weakly separated set-systems. Adv. Math. 224, 1-44 (2010)
5. Elnitsky, S.: Rhombic tilings of polygons and classes of reduced words in Coxeter groups. J. Combin. Theory, Ser. A 77, 193-221 (1997)
6. Fan, C.K.: A Hecke algebra quotient and some combinatorial applications. J. Algebraic Combin. 5, 175-189 (1996)
7. Knuth, D.E.: Axioms and Hulls. Lecture Notes in Computer Science, vol.
606. Springer, Berlin (1992)
8. Leclerc, B., Zelevinsky, A.: Quasicommuting families of quantum Plücker coordinates. Am. Math.
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