A Determinantal Formula for Supersymmetric Schur Polynomials
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چکیده
We derive a new formula for the supersymmetric Schur polynomial sλ(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 sλ(x/y). This new expression gives rise to a determinantal formula for sλ(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.
منابع مشابه
On Characters and Dimension Formulas for Representations of the Lie Superalgebra
We derive a new expression for the supersymmetric Schur polynomials sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m|n) and gives rise to a determinantal formula for sλ(x/y). In the second part, we use this determinantal formula to derive new expressions for the dimension and superdimension of covariant representations Vλ of the Lie superalgebr...
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