Some Orthogonal Polynomials in Four Variables

TL;DR

This paper constructs an orthogonal basis of four-variable polynomials using group symmetries and explores their connection to a quantum Calogero-Sutherland model of four particles.

Contribution

It introduces a new orthogonal polynomial basis in four variables based on group isomorphisms and links it to a quantum many-body system.

Findings

Orthogonal basis of 4-variable polynomials with 2 parameters constructed

Connection established between polynomials and quantum Calogero-Sutherland model

Utilizes symmetry group isomorphisms to develop polynomial systems

Abstract

The symmetric group on 4 letters has the reflection group $D_{3}$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_{3}$ and $D_{3}$. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical particles on the line.