On Dimension Formulas for gl(m | n) Representations<!-- Anfang Autor -->
Journal of Lie Theory, Volume 14 (2004) no. 2, pp. 523-535
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.
@article{JOLT_2004_14_2_a9,
author = {E. M. Moens and J. Van der Jeugt},
title = {On {Dimension} {Formulas} for gl(m | n) {Representations<!--} {Anfang} {Autor} -->},
journal = {Journal of Lie Theory},
pages = {523--535},
year = {2004},
volume = {14},
number = {2},
doi = {10.5802/jolt.349},
zbl = {1085.17007},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.349/}
}
E. M. Moens; J. Van der Jeugt. On Dimension Formulas for gl(m | n) Representations. Journal of Lie Theory, Volume 14 (2004) no. 2, pp. 523-535. doi: 10.5802/jolt.349
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