Pattern-of-zeros approach to Fractional quantum Hall states and a classification of symmetric polynomial of infinite variables

Some purely chiral fractional quantum Hall states are described by symmetric or anti-symmetric polynomials of infinite variables. In this article, we review a systematic construction and classification of those fractional quantum Hall states and the corresponding polynomials of infinite variables, using the pattern-of-zeros approach. We discuss how to use patterns of zeros to label different fractional quantum Hall states and the corresponding polynomials. We also discuss how to calculate various universal properties (ie the quantum topological invariants) from the pattern of zeros.