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  • Proceedings of the Edinburgh Mathematical Society, Volume 43
  • June 2000, pp. 379-393

On Turnbull identity for skew-symmetric matrices

  • Tôru Umeda (a1) and Takeshi Hirai
  • DOI: http://dx.doi.org/10.1017/S0013091500020988
  • Published online: 20 January 2009
Abstract
Abstract

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.

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1. A. Capelli , Über die Zurückführung der Cayley'sehen Operation Ω auf gewöhnliche Polar-Operationen, Math. Ann. 29 (1887), 331338.

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14. M. Noumi , T. Umeda and M. Wakayama , A quantum analogue of the Capelli identity and an elementary differential calculus on GLq(n), Duke Math. J. 76 (1994), 567594.

15. A. Oknounkov , Quantum immanants and higher Capelli identities. Transformation Groups 1 (1996), 99126.


19. T. Umeda , Newton's formula for gln, Proc. Am. Math. Soc. 126 (1998), 31693175.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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