A decomposition of elements of the free algebra

Abstract:

Let $f$ be an element of $F[X]$, the free associative algebra over a field $F$ and $n$ the maximum of the degrees of the variables and the multiplicities of the degrees in $f$. A partial ordering on the homogeneous elements of $F[X]$ is defined such that if $f$ is homogeneous and $F\nmid n!$, then $f$ can be decomposed into a sum of two polynomials ${f_0}$ and ${f_1}$ such that for $0 < m \leqslant n,\;{f_0}$ is symmetric or skew symmetric in all its arguments of degree $m$ depending on whether $m$ is even or odd and ${f_1}$ is a consequence of polynomials of lower type than $f$. Osborn’s Theorem about the symmetry of the absolutely irreducible polynomial identities is obtained as a corollary. The same holds in the free nonassociative algebra. The proofs are combinatorial.