On Differentiating Symmetric Functions

TL;DR

This paper investigates special symmetric polynomials that simplify dual differential operators, revealing elegant algebraic structures and connections to Bell polynomials, with applications to computing gradients of symmetric functions.

Contribution

It identifies symmetric polynomials with simplified dual differential operators and explores their properties and relations to Weyl algebra and Bell polynomials.

Findings

Derived simple forms for dual differential operators in symmetric polynomials

Connected symmetric polynomial coordinates to Bell polynomials

Provided formulas for gradients of symmetric functions at arbitrary points

Abstract

A symmetric function of $N$ variables can be given in terms of symmetric polynomials of these variables. We determine those symmetric polynomials in which the dual differential operators take the neatest form when expressed in terms of our original variables, producing a simple form for the associated Weyl algebra. Both our coordinates, and the form of our differential operators at total diagonal points, exhibit interesting properties, and are related to interesting objects like Bell polynomials. They can be modified to give a simple formula for the gradient of our symmetric function at any point.