A Determinantal Formula for Supersymmetric Schur Polynomials |
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Authors: | EM Moens J Van der Jeugt |
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Institution: | (1) Department of Applied Mathematics and Computer Science, University of Ghent, Krijgslaan 281-S9, B-9000 Gent, Belgium |
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Abstract: | We derive a new formula for the supersymmetric Schur polynomial s
(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
(x/y). This new expression gives rise to a determinantal formula for s
(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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Keywords: | supersymmetric Schur polynomials Lie superalgebra gl(m/n) characters covariant tensor representations determinantal identities |
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