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Mathematical Physics Seminar

In-Person Seminar: Bhargav Narayanan - Elementary symmetric polynomials under the fixed point measure

Location:  Hill 705
Date & time: Thursday, 16 October 2025 at 12:10PM - 1:10PM

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.

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