Character and determinant formulae of quasifinite representation of the W$_{1+\infty}$ algebra

摘要

We diagonalize the Hilbert space of the quasifinite module of the \Winf algebra. States are classified according to their eigenvalues for infinitely many commuting charges and the Young diagrams. The parameter dependence of their norms is explicitly derived. The full character formulae of the degenerate representations are given as summation of the bilinear combinations of the Schur polynomials.