A $q$-analogue of the FKG inequality and some applications

Anders Björner

Published 2009-06-07, updated 2009-08-21Version 3

Let $L$ be a finite distributive lattice and $\mu : L \to {\mathbb R}^{+}$ a log-supermodular function. For functions $k: L \to {\mathbb R}^{+}$ let $$E_{\mu} (k; q) \defeq \sum_{x\in L} k(x) \mu (x) q^{{\mathrm rank}(x)} \in {\mathbb R}^{+}[q].$$ We prove for any pair $g,h: L\to {\mathbb R}^{+}$ 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 ``$ \ll $'' 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 $k, \mu: Y \rarr \R^+$, and parameters $0\le s\le t$, define the formal power series $$F_{\mu}(k ; z) \defeq \sum_{\la\in Y} k(\la) \mu(\la) (f_{\la})^t \frac{z^{|\la|}}{(|\la| !)^s} \in \R^+ [[z]], $$ %\sum_{\la\in Y} k(\la) \mu(\la) (f_{\la})^t \frac{z^{|\la|}}{|\la| !} \in \R^+ [[z]],$$ where $f_{\la}$ is the number of standard Young tableaux of shape $\la$. Assume that $\mu: Y\rarr \R^+$ is log-supermodular, and that $g, h: Y \rarr \R^+$ are monotonely increasing with respect to containment order of partition shapes. Then $$F_{\mu}(g;z) \cdot F_{\mu}(h;z) \ll F_{\mu}(1;z) \cdot F_{\mu}(gh;z). $$