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- W2950257557 abstract "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 $$E_{mu} (k; q) defeq sum_{xin L} k(x) mu (x) q^{{mathrm rank}(x)} in {mathbb R}^{+}[q].$$ We prove for any pair $g,h: Lto {mathbb R}^{+}$ 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 $0le sle t$, define the formal power series $$F_{mu}(k ; z) defeq sum_{lain Y} k(la) mu(la) (f_{la})^t frac{z^{|la|}}{(|la| !)^s} in R^+ [[z]], $$ %sum_{lain Y} k(la) mu(la) (f_{la})^t frac{z^{|la|}}{|la| !} in R^+ [[z]],$$ where $f_{la}$ is the number of standard Young tableaux of shape $la$. Assume that $mu: Yrarr R^+$ is log-supermodular, and that $g, h: Y rarr R^+$ are monotonely increasing with respect to containment order of partition shapes. Then $$F_{mu}(g;z) cdot F_{mu}(h;z) ll F_{mu}(1;z) cdot F_{mu}(gh;z). $$" @default.
- W2950257557 created "2019-06-27" @default.
- W2950257557 creator A5024148075 @default.
- W2950257557 date "2009-06-07" @default.
- W2950257557 modified "2023-09-27" @default.
- W2950257557 title "A $q$-analogue of the FKG inequality and some applications" @default.
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