Bhargav Narayanan – Rutgers University
Date/Time/Location
Thursday, October 16th, 2025, 12:10 pm; Hill Center 705
Elementary symmetric polynomials under the fixed point measure
I’ll talk about a surprising inequality satisfied by elementary symmetric polynomials under the action of the fixed point measure of a random permutation. Concretely, Ayush Khaita, Ishan Mata and I recently proved that any n non-negative real numbers a_1, a_2, dots, a_n satisfy the inequality
(1/n!) sum_{f in S_n} prod_{i:i=f(i)} a_i
> =
(1/(n choose 2)) sum_{{j,k} in ([n] choose 2)} sqrt{a_j a_k}.
This bound is sharp, and equality is attained if and only if a_i = 1 for all 1 le i le n.
Where does this inequality come from? This is arguably much more interesting than our proof itself, as I’ll aim to convince you in my talk.