Combinatorica

, 31:151 | Cite as

A q-analogue of the FKG inequality and some applications

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Abstract

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 

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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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