Explicit computations with the Divided Symmetrization operator

TL;DR

This paper derives explicit closed-form formulas for the divided symmetrization of various polynomials, facilitating computations of symmetric functions and related geometric volumes, with simple proof methods applicable to broader cases.

Contribution

It provides new closed-form divided symmetrization formulas for different polynomials, expanding computational tools in symmetric function theory and polytope volume calculations.

Findings

Closed-form DS formulas for multiple polynomials

Simplified proof techniques for DS computations

Applications to permutohedron volume calculations

Abstract

Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable and available to the research practitioner in these areas. Also, the methods of proof utilized here are simple and amenable to many more analogous computations. We conclude the paper with a list of such formulas.