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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Newton’s formula for $\mathfrak {gl}_{n}$
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by Tôru Umeda PDF
Proc. Amer. Math. Soc. 126 (1998), 3169-3175 Request permission

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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Additional Information
  • Tôru Umeda
  • Affiliation: Department of Mathematics, Kyoto University, Kyoto 606, Japan
  • Email: umeda@kusm.kyoto-u.ac.jp
  • Received by editor(s): March 28, 1997

  • Dedicated: Dedicated to Professor Reiji Takahashi on the occasion of his seventieth birthday
  • Communicated by: Roe Goodman
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3169-3175
  • MSC (1991): Primary 17B35, 15A33
  • DOI: https://doi.org/10.1090/S0002-9939-98-04557-2
  • MathSciNet review: 1468206