, 31:151 | Cite as

A q-analogue of the FKG inequality and some applications

  • Anders Björner


Let L be a finite distributive lattice and µ: L → ℝ+ a log-supermodular function. For functions k: L → ℝ+ let
$$E_\mu (k;q)^{\underline{\underline {def}} } \sum\limits_{x \in L} {k(x)\mu (x)q^{rank(x)} \in \mathbb{R}^ + [q]} .$$
We prove for any pair g,h: L → ℝ+ of monotonely increasing functions, that
$$E_\mu (g;q) \cdot E_\mu (h;q) \ll E_\mu (1;q) \cdot E_\mu (gh;q),$$
where “≪” denotes coefficientwise inequality of real polynomials. The FKG inequality of Fortuin, Kasteleyn and Ginibre (1971) is the real number inequality obtained by specializing to q=1.

The polynomial FKG inequality has applications to f-vectors of joins and intersections of simplicial complexes, to Betti numbers of intersections of Schubert varieties, and to correlation-type inequalities for a class of power series weighted by Young tableaux. This class contains series involving Plancherel measure for the symmetric groups and its poissonization.

Mathematics Subject Classification (2000)

05A20 05E10 60C05 


  1. [1]
    R. Ahlswede and D. E. Daykin: An inequality for the weights of two families of sets, their unions and intersections; Z. Wahrs. Verw. Gebiete 43 (1978), 183–185.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    N. Alon and J. H. Spencer: The Probabilistic Method (3rd ed.), Wiley-Interscience, New York, 2008.zbMATHCrossRefGoogle Scholar
  3. [3]
    S. Billey and V. Lakshmibai: Singular Loci of Schubert Varieties, Progress in Math. No. 182, Birkhäuser, Boston, 2000.Google Scholar
  4. [4]
    G. Birkhoff: Lattice Theory (3rd ed.), Amer.Math. Soc. Colloquium Publ. No. 25, American Mathematical Society, Providence, RI, 1967.zbMATHGoogle Scholar
  5. [5]
    A. Borodin, A. Okounkov and G. Olshanski: Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), 481–515.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    G. R. Brightwell and W. T. Trotter: A combinatorial approach to correlation inequalities, Discrete Math. 257 (2002), 311–327.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    D. Christofides: A q-analogue of the four functions theorem, Combinatorica, to appear.Google Scholar
  8. [8]
    P. C. Fishburn: A correlational inequality for linear extensions of a poset, Order 1 (1984), 127–137.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    C. M. Fortuin, P. W. Kasteleyn and J. Ginibre: Corrrelation inequalities on some partially ordered sets, Commun. Math. Phys. 22 (1971), 89–103.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    H. Hiller: Geometry of Coxeter Groups, Research Notes in Math. No. 54, Pitman, Boston, 1982.zbMATHGoogle Scholar
  11. [11]
    D. J. Kleitman: Families of non-disjoint subsets, J. Combinat. Theory 1 (1966), 153–155.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    R. A. Proctor: Bruhat lattices, plane partition generating functions, and minuscule representations; Europ. J. Combinatorics 5 (1984), 331–350.MathSciNetzbMATHGoogle Scholar
  13. [13]
    S. Sahi: The FKG inequality for partially ordered algebras, J. Theor. Probab. 21 (2008), 449–458.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    R. P. Stanley: Enumerative Combinatorics, Vol. 1; Cambridge Studies in Advanced Mathemtics No. 49, Cambridge University Press, Cambridge, UK, 1997.zbMATHGoogle Scholar
  15. [15]
    R. P. Stanley: Enumerative Combinatorics, Vol. 2; Cambridge Studies in Advanced Mathemtics No. 62, Cambridge University Press, Cambridge, UK, 1999.CrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  1. 1.Institut Mittag-LefflerDjursholmSweden
  2. 2.Kungl. Tekniska HögskolanMatematiska Inst.StockholmSweden

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