Hostname: page-component-cb9f654ff-hn9fh Total loading time: 0 Render date: 2025-08-19T08:37:32.487Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Information

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

References

1. Capelli, A., Über die Zurückführung der Cayley'sehen Operation Ω auf gewöhnliche Polar-Operationen, Math. Ann. 29 (1887), 331–338.CrossRefGoogle Scholar
2. Capelli, A., Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37 (1890), 1–37.CrossRefGoogle Scholar
3. Foata, D. and Zeilberger, D., Combinatorial proofs of Capelli's and Turnbull's identities from classical invariant theory, Electron. J. Combin. 1 (1994), 10pp.CrossRefGoogle Scholar
4. Howe, R., Remarks on classical invariant theory, Trans. Am. Math. Soc. 313 (1989), 539–570. (Erratum: Trans. Am. Math. Soc. 318 (1990), 823.)CrossRefGoogle Scholar
5. Howe, R. and Umeda, T., The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), 565–619.CrossRefGoogle Scholar
6. Itoh, M. and Umeda, T., On central elements in the universal enveloping algebras of the orthogonal Lie algebras, preprint, 1998.Google Scholar
7. Kostant, B. and Sahi, S., The Capelli identity, tube domains and the generalized Laplace transform, Adv. Math. 87 (1991), 71–92.CrossRefGoogle Scholar
8. Koszul, J.-L., Les algèbre de Lie graduée de type sl(n, 1) et l'opérateur de A. Capelli, C. R. Acad. Sci. Paris 292 (1981), 139–141.Google Scholar
9. Meyer, F., Bericht über den gegenwärtigen Stand der Invariantentheorie, Jber. d. Dt. Math.-Verein 1 (1892), 79–292.Google Scholar
10. Molev, A. and Nazarov, M., Capelli identities for classical Lie algebras, Math. Ann. 313 (1999), 315–357.CrossRefGoogle Scholar
11. Molev, A., Nazarov, M. and Olshanskii, G., Yangians and classical Lie algebras, Russ. Math. Surv. 51 (1996), 205–282.CrossRefGoogle Scholar
12. Nazarov, M., Quantum Berezinian and the classical Capelli identity, Lett. Math. Phys. 21 (1991), 123–131.CrossRefGoogle Scholar
13. Nazarov, M., Yangians and Capelli identities, in Kirillov's seminar on representation theory (ed. Olshanski, G. I.), AMS Translations, Series 2, vol. 181 (1998), pp. 139–163.Google Scholar
14. Noumi, M., Umeda, T. and Wakayama, M., A quantum analogue of the Capelli identity and an elementary differential calculus on GLq(n), Duke Math. J. 76 (1994), 567–594.CrossRefGoogle Scholar
15. Oknounkov, A., Quantum immanants and higher Capelli identities. Transformation Groups 1 (1996), 99–126.CrossRefGoogle Scholar
16. Turnbull, H. W., The theory of determinants, matrices, and invariants (Dover, London, 1960).Google Scholar
17. Turnbull, H. W., Symmetric determinants and the Cayley and Capelli operators Proc. Edinb. Math. Soc. 8 (1948), 76–86.CrossRefGoogle Scholar
18. Umeda, T., The Capelli identities, a century after, Sugaku 46 (1994), 206–227 (in Japanese). (English translation: Selected Papers on Harmonic Analysis, Groups, and Invariants (ed. Nomizu, K.), AMS Translations, Series 2, vol. 183 (1998), pp. 51–78.)Google Scholar
19. Umeda, T., Newton's formula for gln, Proc. Am. Math. Soc. 126 (1998), 3169–3175.CrossRefGoogle Scholar
20. Umeda, T., On the proof of the Capelli identities, preprint, 1997.CrossRefGoogle Scholar
21. Weyl, H., The classical groups, their invariants and representations (Princeton University Press, Princeton, NJ, 1946).Google Scholar
22. Želobenko, D. P., Compact Lie groups and their representations, Translations of Mathematical Monographs, vol. 40 (American Mathematical Society, 1973).CrossRefGoogle Scholar