A q-analogue of the FKG inequality and some applications

Let L be a finite distributive lattice and μ : L → R a logsupermodular function. For functions k : L → R let Eμ(k; q) def = ∑ x∈L k(x)μ(x)q ∈ R[q]. We prove for any pair g, h : L → R of monotonely increasing functions, that Eμ(g; q) ·Eμ(h; q) ≪ Eμ(1; q) · Eμ(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 certain Schubert varieties, and to the following kind of correlation inequality for power series weighted by Young tableaux. Let Y be the set of all integer partitions. Given functions k, μ : Y → R, define the formal power series Fμ(k; z) def = ∑ λ∈Y k(λ)μ(λ) fλ z |λ|! ∈ R[[z]], where fλ is the number of standard Young tableaux of shape λ. Assume that μ : Y → R is log-supermodular, and that g, h : Y → R are monotonely increasing with respect to containment order of partition shapes. Then Fμ(g; z) · Fμ(h; z) ≪ Fμ(1; z) · Fμ(gh; z).