c © 2004 Heldermann Verlag On Dimension Formulas for gl(m|n) Representations

Abstract. We investigate new formulas for the dimension and superdimen-sion of covariant representations Vλ of the Lie superalgebra gl(m|n). The notion of t-dimension is introduced, where the parameter t keeps track of the Z-grading of Vλ. Thus when t = 1, the t-dimension reduces to the ordinary dimension, and when t = −1 it reduces to the superdimension. An interesting formula for the t-dimension is derived from a recently obtained new formula for the su-persymmetric Schur polynomial sλ(x/y), which yields the character of Vλ. It expresses the t-dimension as a simple determinant. For a special choice of λ, the new t-dimension formula gives rise to a Hankel determinant identity. 1