Standard Polynomials in Matrix Algebras

Let be an <!-- MATH $n \times n$ --> matrix ring with entries in the field F, and let <!-- MATH ${S_k}({X_1}, \ldots ,{X_k})$ --> be the standard polynomial in k variables. Amitsur-Levitzki have shown that <!-- MATH ${S_{2n}}({X_1}, \ldots ,{X_{2n}})$ --> vanishes for all specializations of <!-- MATH ${X_1}, \ldots ,{X_{2n}}$ --> to elements of . Now, with respect to the transpose, let <!-- MATH $M_n^ - (F)$ --> be the set of antisymmetric elements and let <!-- MATH $M_n^ + (F)$ --> be the set of symmetric elements. Kostant has shown using Lie group theory that for n even <!-- MATH ${S_{2n - 2}}({X_1}, \ldots ,{X_{2n - 2}})$ --> vanishes for all specializations of <!-- MATH ${X_1}, \ldots ,{X_{2n - 2}}$ --> to elements of <!-- MATH $M_n^ - (F)$ --> . By strictly elementary methods we have obtained the following strengthening of Kostant's theorem: