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.