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 FSn, of the symmetric group S-n, over a field F of characteristic 0 (or p > n). The goal is to obtain a constructive version of the isomorphism psi: circle plus(lambda) Md-lambda (F) -> FSn where lambda is a partition of n and d(lambda) counts the standard tableaux of shape lambda. Young showed how to compute psi; to compute its inverse, we use an efficient algorithm for representation ;matrices discovered by Clifton. In 2, we discuss constructive methods based on 1 which allow us to analyze the polynomial identities satisfied by a specific (non)associative algebra: fill and reduce algorithm, module generators algorithm, Bondari's algorithm for finite dimensional algebras. In 3, we study the multilinear identities satisfied by the octonion algebra O over a field of characteristic 0. For n <= 6 we compare our computational results with earlier work of Racine, Hentzel & Peresi, Shestakov & Zhukavets. Going one step further, we verify computationally that every identity in degree 7 is a consequence of known identities of lower degree; this result is our main original contribution. This gap (no new identities in degree 7) motivates our concluding conjecture: the known identities for n <= 6 generate all of the octonion identities in characteristic 0.