Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

On Turnbull identity for skew-symmetric matrices

Tôru Umedaa1 and Professor Takeshi Hirai

a1 Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan


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.

(Received August 20 1997)


  • centre of universal enveloping algebra;
  • Capelli identity;
  • permanent