### Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions

This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $\mathbb{F}{S}_{n}$ of the symmetric group ${S}_{n}$ over a field $\mathbb{F}$ of characteristic 0 (or $p>n$). The goal is to obtain a constructive version of the isomorphism $\psi :{\u2a01}_{\lambda}{M}_{{d}_{\lambda}}\left(\mathbb{F}\right)\u27f6\mathbb{F}{S}_{n}$ where $\lambda $ is a partition of $n$ and ${d}_{\lambda}$ counts the standard tableaux of shape $\lambda $....