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Some purely chiral fractional quantum Hall states are described by symmetricor anti-symmetric polynomials of infinite variables. In this article, we reviewa systematic construction and classification of those fractional quantum Hallstates and the corresponding polynomials of infinite variables, using thepattern-of-zeros approach. We discuss how to use patterns of zeros to labeldifferent fractional quantum Hall states and the corresponding polynomials. Wealso discuss how to calculate various universal properties (ie the quantumtopological invariants) from the pattern of zeros.

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