
Journal of Lie Theory 14 (2004), No. 2, 523535 Copyright Heldermann Verlag 2004 On Dimension Formulas for gl(m  n) Representations E. M. Moens Dept. of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281S9, 9000 Gent, Belgium, ElsM.Moens@ugent.be Joris Van der Jeugt Dept. of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281S9, 9000 Gent, Belgium, Joris.VanderJeugt@ugent.be [Abstractpdf] We investigate new formulas for the dimension and superdimension of covariant representations $V_\lambda$ of the Lie superalgebra $\frak{gl}(m{}n)$. The notion of $t$dimension is introduced, where the parameter $t$ keeps track of the $\mathbb Z$grading of $V_\lambda$. 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 supersymmetric Schur polynomial $s_\lambda(x/y)$, which yields the character of $V_\lambda$. It expresses the $t$dimension as a simple determinant. For a special choice of $\lambda$, the new $t$dimension formula gives rise to a Hankel determinant identity. [FullTextpdf (192 KB)] for subscribers only. 