## 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.

## References

- A. S. Amitsur and J. Levitzki,
*Minimal identities for algebras*, Proc. Amer. Math. Soc.**1**(1950), 449–463. MR**36751**, DOI 10.1090/S0002-9939-1950-0036751-9 - Nathan Jacobson,
*Structure of rings*, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR**0222106** - Bertram Kostant,
*A theorem of Frobenius, a theorem of Amitsur-Levitski and cohomology theory*, J. Math. Mech.**7**(1958), 237–264. MR**0092755**, DOI 10.1512/iumj.1958.7.57019 - Oystein Ore,
*Theory of graphs*, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R.I., 1962. MR**0150753** - Frank W. Owens,
*Applications of graph theory to matrix theory*, Proc. Amer. Math. Soc.**51**(1975), 242–249. MR**376708**, DOI 10.1090/S0002-9939-1975-0376708-9
L. Rowen, Thesis, Yale University, New Haven, Conn., 1973.

## 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