Excited States of Calogero-Sutherland Model and Singular Vectors of the $W_N$ Algebra

Abstract

Using the collective field method, we find a relation between the Jack symmetric polynomials, which describe the excited states of the Calogero-Sutherland model, and the singular vectors of the $W_N$ algebra. Based on this relation, we obtain their integral representations. We also give a direct algebraic method which leads to the same result, and integral representations of the skew-Jack polynomials.

LaTeX, 29 pages, 2 figures, New sections for skew-Jack polynomial and example of singular vectors added