Standard polynomials in matrix algebras

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.