Mathematics – Combinatorics
Scientific paper
2009-06-07
Mathematics
Combinatorics
Version 2: minor corrections not affecting math Version 3: improved presentation of the results in Sections 5 and 6
Scientific paper
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). $$
No associations
LandOfFree
A $q$-analogue of the FKG inequality and some applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A $q$-analogue of the FKG inequality and some applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A $q$-analogue of the FKG inequality and some applications will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-231503