Abstract
We derive a new formula for the supersymmetric Schur polynomial s λ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for s λ(x/y). This new expression gives rise to a determinantal formula for s λ(x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.
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Moens, E., Van der Jeugt, J. A Determinantal Formula for Supersymmetric Schur Polynomials. Journal of Algebraic Combinatorics 17, 283–307 (2003). https://doi.org/10.1023/A:1025048821756
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DOI: https://doi.org/10.1023/A:1025048821756