On translation invariant symmetric polynomials and Haldane's conjecture

TL;DR

This paper proves that the ring of translation invariant symmetric polynomials is isomorphic to a polynomial ring in fewer variables and disprove Haldane's conjecture, with implications for quantum Hall wavefunctions.

Contribution

It establishes an isomorphism for translation invariant symmetric polynomials and refutes Haldane's conjecture, advancing understanding in mathematical physics.

Findings

Ring of translation invariant symmetric polynomials is isomorphic to a polynomial ring in n-1 variables

Disproof of Haldane's conjecture regarding polynomial structure

Implications for fractional quantum Hall effect wavefunctions

Abstract

We show that the ring of translation invariant symmetric polynomials in n variables is isomorphic to the full polynomial ring in n-1 variables, in characteristic 0. We disprove a conjecture of Haldane regarding the structure of such polynomials. Our motivation is the fractional quantum Hall effect, where translation invariant (anti)symmetric complex n-variate polynomials characterize n-electron wavefunctions.