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Transactions of the American Mathematical Society

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Standard polynomials in matrix algebras


Author: Louis H. Rowen
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
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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.


References [Enhancements On Off] (What's this?)

  • [1] S. A. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449-463. MR 12, 155. MR 0036751 (12:155d)
  • [2] N. Jacobson, Structure of rings, 2nd rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R.I., 1964, Chap. X. MR 36 #5158. MR 0222106 (36:5158)
  • [3] B. Kostant, A theorem of Frobenius, a theorem of Amitsur-Levitzki, and cohomology theory, J. Math. Mech. 7 (1958), 237-264. MR 19, 1153. MR 0092755 (19:1153e)
  • [4] Oystein Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ., vol. 38, Amer. Math. Soc., Providence, R.I., 1962. MR 27 #740. MR 0150753 (27:740)
  • [5] F. Owens, Applications of graph theory to matrix theory (to appear). MR 0376708 (51:12883)
  • [6] L. Rowen, Thesis, Yale University, New Haven, Conn., 1973.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0349715-3
Keywords: Antisymmetric, involution, matrix algebra, polynomial identity, standard identity, symmetric, transpose
Article copyright: © Copyright 1974 American Mathematical Society

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