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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Standard polynomials in matrix algebras
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by Louis H. Rowen PDF
Trans. Amer. Math. Soc. 190 (1974), 253-284 Request permission

Abstract:

Let ${M_n}(F)$ be an $n \times n$ matrix ring with entries in the field F, and let ${S_k}({X_1}, \ldots ,{X_k})$ be the standard polynomial in k variables. Amitsur-Levitzki have shown that ${S_{2n}}({X_1}, \ldots ,{X_{2n}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n}}$ to elements of ${M_n}(F)$. Now, with respect to the transpose, let $M_n^ - (F)$ be the set of antisymmetric elements and let $M_n^ + (F)$ be the set of symmetric elements. Kostant has shown using Lie group theory that for n even ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$. By strictly elementary methods we have obtained the following strengthening of Kostant’s theorem: ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$, for all n. ${S_{2n - 1}}({X_1}, \ldots ,{X_{2n - 1}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 2}}$ to elements of $M_n^ - (F)$ and of ${X_{2n - 1}}$ to an element of $M_n^ + (F)$, for all n. ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ vanishes for all specializations of ${X_1}, \ldots ,{X_{2n - 3}}$ to elements of $M_n^ - (F)$ and of ${X_{2n - 2}}$ to an element of $M_n^ + (F)$, for n odd. These are the best possible results if F has characteristic 0; a complete analysis of the problem is also given if F has characteristic 2.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 190 (1974), 253-284
  • MSC: Primary 15A30
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0349715-3
  • MathSciNet review: 0349715