Multilinear identities of the matrix ring

Abstract:

Let V be a vector space over a field F of zero characteristic, which is acted upon by the symmetric group. Systems of generators for V are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree $2n + 1$ which holds in the matrix ring ${F_n}(n > 2)$ is a consequence of the standard identity ${s_{2n}}$. The notions of rigid and semirigid sequences of matrices are defined and treated.