Classification of symmetric polynomials of infinite variables: Construction of Abelian and non-Abelian quantum Hall states

The classification of complex wave functions of infinite variables is an important problem since it is related to the classification of possible quantum states of matter. In this paper, we propose a way to classify symmetric polynomials of infinite variables using the pattern of zeros of the polynomials. Such a classification leads to a construction of a class of simple non-Abelian quantum Hall states which are closely related to parafermion conformal field theories.