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Journal of Algebraic Combinatorics

, Volume 17, Issue 3, pp 283–307 | Cite as

A Determinantal Formula for Supersymmetric Schur Polynomials

  • E.M. Moens
  • J. Van der Jeugt
Article

Abstract

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.

supersymmetric Schur polynomials Lie superalgebra gl(m/ncharacters covariant tensor representations determinantal identities 

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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • E.M. Moens
    • 1
  • J. Van der Jeugt
    • 1
  1. 1.Department of Applied Mathematics and Computer ScienceUniversity of GhentGentBelgium

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