Elementary symmetric polynomials under the fixed point measure

TL;DR

This paper proves a sharp inequality involving elementary symmetric polynomials under the fixed point measure of a permutation, using differential operators and a monotone flow to establish the result.

Contribution

The paper introduces a novel inequality for elementary symmetric polynomials under permutation fixed point measures and develops a differential operator-based method to prove it.

Findings

Established a sharp inequality for elementary symmetric polynomials under fixed point measure.

Developed a differential operator and monotone flow approach for proving inequalities.

Provided a new technique applicable to symmetric polynomial inequalities.

Abstract

We identify a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, for any collection of $n$ non-negative real numbers $a_1, \dots, a_n \in \mathbb{R}_{\geq 0}$, we prove that \[ \frac{1}{n!} \sum_{\pi \in S_n} \left[\prod_{\{i:i=\pi(i)\}} a_i\right] \ge \frac{1}{\binom{n}{2}} \sum_{S \in\binom{[n]}{2}} \left[ \left(\prod_{\{i \in S\}} a_i \right)^{1/2}\right], \] and this bound is sharp. To prove this elementary inequality, we construct a collection of differential operators to set up a monotone flow that then allows us to establish the inequality.