A q-analogue of the FKG inequality and some applications

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.

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Correspondence to Anders Björner.

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Research supported by the Knut and Alice Wallenberg Foundation, grant KAW.2005.0098.

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Björner, A. A q-analogue of the FKG inequality and some applications. Combinatorica 31, 151 (2011). https://doi.org/10.1007/s00493-011-2644-1

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Mathematics Subject Classification (2000)

  • 05A20
  • 05E10
  • 60C05