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On Turnbull identity for skew-symmetric matrices

Published online by Cambridge University Press:  20 January 2009

Tôru Umeda  and
Takeshi Hirai
Tôru Umeda
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
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Abstract

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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.

Keywords

centre of universal enveloping algebraCapelli identitypermanent
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 43 , Issue 2 , June 2000 , pp. 379 - 393
Copyright
Copyright © Edinburgh Mathematical Society 2000

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