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2 results

  • On Turnbull identity for skew-symmetric matrices

  • Tôru Umeda, Takeshi Hirai
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 43 / Issue 2 / June 2000
    Published online by Cambridge University Press:
    20 January 2009, pp. 379-393
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  • In the last six lines of Turnbull's 1948 paper, he left an enigmatic statement on a Capelli-type identity for skew-symmetric matrix spaces. In the present paper, on Turnbull's suggestion, we show that certain Capelli-type identities hold for this case. Our formulae connect explicitly the central elements in U(gln) to the invariant differential operators, both of which are expressed with permanent. This also clarifies the meaning of Turnbull's statement from the Lie-theoretic point of view.

  • EXPLICIT CAPELLI IDENTITIES FOR SKEW SYMMETRIC MATRICES

  • Kenji Kinoshita, Masato Wakayama
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 45 / Issue 2 / June 2002
    Published online by Cambridge University Press:
    17 June 2002, pp. 449-465
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  • The main purpose of this paper is to obtain an explicit Capelli identity relating skew-symmetric matrices under the action of the general linear group $GL_N$. In particular, we give an explicit formula for the skew Capelli element in terms of the trace of powers of a matrix defined by the standard infinitesimal generators of $GL_N$.

    AMS 2000 Mathematics subject classification: Primary 17B35. Secondary 17B10; 15A15; 20G05