On Dimension Formulas for $\g\l(m{|}n)$ Representations

E. M. Moens and J. Van der Jeugt

Abstract: We investigate new formulas for the dimension and superdimension of covariant representations $V_\lambda$ of the Lie superalgebra $\g\l(m{|}n)$. The notion of $t$-dimension is introduced, where the parameter $t$ keeps track of the $\Z$-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.

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