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

Abstract

In part 1, we review the structure theory of $\mathbb{F} S_n$, the group algebra of the symmetric group $S_n$ over a field of characteristic 0. We define the images $ψ(E^λ_{ij})$ of the matrix units $E^λ_{ij}$ ($1 \le i, j \le d_λ$), where $d_λ$ is the number of standard tableaux of shape $λ$, and obtain an explicit construction of Young's isomorphism $ψ\colon \bigoplus_λM_{d_λ}(\mathbb{F}) \to \mathbb{F} S_n$. We then present Clifton's algorithm for the construction of the representation matrices $R^λ(p) \in M_{d_λ}(\mathbb{F})$ for all $p \in S_n$, and obtain the reverse isomorphism $ϕ\colon \mathbb{F} S_n \to \bigoplus_λM_{d_λ}(\mathbb{F})$. In part 2, we apply the structure theory of $\mathbb{F} S_n$ to the study of multilinear polynomial identities of degree $n \le 7$ for the algebra $\mathbb{O}$ of octonions over a field of characteristic 0. We compare our results with earlier work of Racine, Hentzel & Peresi, and Shestakov & Zhukavets on the identities of degree $n \le 6$. We use computational linear algebra to verify that every identity in degree 7 is a consequence of the known identities of lower degrees: there are no new identities in degree 7. We conjecture that the known identities of degree $\le 6$ generate all octonion identities in characteristic 0.