On Translation Invariant Symmetric Polynomials and Haldane's Conjecture

Abstract

We establish an isomorphism between the ring of translation invariant symmetric polynomials in n variables and the full polynomial ring in n − 1 variables, over any field of characteristic 0. In addition, we give a counterexample to a conjecture of Haldane3 regarding the structure of translation invariant symmetric polynomials. Our motivation is the fractional quantum Hall effect, where translation invariant (anti)symmetric complex polynomials in n variables characterize n-electron wavefunctions.