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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Newton's formula for $\mathfrak{gl}_{n}$


Author: Tôru Umeda
Journal: Proc. Amer. Math. Soc. 126 (1998), 3169-3175
MSC (1991): Primary 17B35, 15A33
MathSciNet review: 1468206
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Abstract: This paper presents an explicit relation between the two sets which are well-known generators of the center of the universal enveloping algebra $U(\mathfrak{gl}_{n})$ of the Lie algebra $\mathfrak{gl}_{n}$: one by Capelli (1890) and the other by Gelfand (1950). Our formula is motivated to give an exact analogy for the classical Newton's formula connecting the elementary symmetric functions and the power sum symmetric functions. The formula itself can be deduced from a more general result on Yangians obtained by Nazarov. Our proof is elementary and has an advantage in its direct accessibility.


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Tôru Umeda
Affiliation: Department of Mathematics, Kyoto University, Kyoto 606, Japan
Email: umeda@kusm.kyoto-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04557-2
PII: S 0002-9939(98)04557-2
Keywords: Center of universal enveloping algebra, Newton's formula, Hamilton-Cayley theorem
Received by editor(s): March 28, 1997
Dedicated: Dedicated to Professor Reiji Takahashi on the occasion of his seventieth birthday
Communicated by: Roe Goodman
Article copyright: © Copyright 1998 American Mathematical Society