Journal of Algebraic Combinatorics

, Volume 32, Issue 4, pp 497–531 | Cite as

On maximal weakly separated set-systems

  • Vladimir I. Danilov
  • Alexander V. Karzanov
  • Gleb A. Koshevoy


For a permutation ωS 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,…,n} and conjectured that all inclusionwise maximal collections of this sort have the same cardinality (ω)+n+1, where (ω) is the length of ω. We answer this conjecture affirmatively and present a generalization and additional results.


Weakly separated sets Rhombus tiling Generalized tiling Weak Bruhat order Cluster algebras 


  1. 1.
    Berenstein, A., Fomin, S., Zelevinsky, A.: Parametrizations of canonical bases and totally positive matrices. Adv. Math. 122, 49–149 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Birkhoff, G.: Lattice Theory, 3rd edn. Am. Math. Soc., Providence (1967) zbMATHGoogle Scholar
  3. 3.
    Danilov, V., Karzanov, A., Koshevoy, G.: Tropical Plücker functions and their bases. In: Litvinov, G.L., Sergeev, S.N. (eds.) Tropical and Idempotent Mathematics. Contemp. Math. 495, 127–158 (2009) Google Scholar
  4. 4.
    Danilov, V., Karzanov, A., Koshevoy, G.: Plücker environments, wiring and tiling diagrams, and weakly separated set-systems. Adv. Math. 224, 1–44 (2010) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Elnitsky, S.: Rhombic tilings of polygons and classes of reduced words in Coxeter groups. J. Combin. Theory, Ser. A 77, 193–221 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fan, C.K.: A Hecke algebra quotient and some combinatorial applications. J. Algebraic Combin. 5, 175–189 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Knuth, D.E.: Axioms and Hulls. Lecture Notes in Computer Science, vol. 606. Springer, Berlin (1992) zbMATHGoogle Scholar
  8. 8.
    Leclerc, B., Zelevinsky, A.: Quasicommuting families of quantum Plücker coordinates. Am. Math. Soc. Transl., Ser. 2 181, 85–108 (1998) MathSciNetGoogle Scholar
  9. 9.
    Stembridge, J.: On the fully commutative elements of Coxeter groups. J. Algebr. Comb. 5, 353–385 (1996) zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Vladimir I. Danilov
    • 1
  • Alexander V. Karzanov
    • 2
  • Gleb A. Koshevoy
    • 1
  1. 1.Central Institute of Economics and Mathematics of the RASMoscowRussia
  2. 2.Institute for System Analysis of the RASMoscowRussia

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