Some Graded Identities of The Cayley-Dickson Algebra

We work to find a basis of graded identities for the octonion algebra. We do so for the $\mathbb{Z}_2^2$ and $\mathbb{Z}_2^3$ gradings, both of them derived of the Cayley-Dickson process, the later grading being possible only when the characteristic of the scalars is not two

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

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 $\psi(E^\lambda_{ij})$ of the matrix units $E^\lambda_{ij}$ ($1 \le i, j \le d_\lambda$), where $d_\lambda$ is the number of standard tableaux of shape $\lambda$, and obtain an explicit construction of Young's isomorphism $\psi\colon \bigoplus_\lambda M_{d_\lambda}(\mathbb{F}) \to \mathbb{F} S_n$. We then present Clifton's algorithm for the construction of the representation matrices $R^\lambda(p) \in M_{d_\lambda}(\mathbb{F})$ for all $p \in S_n$, and obtain the reverse isomorphism $\phi\colon \mathbb{F} S_n \to \bigoplus_\lambda M_{d_\lambda}(\mathbb{F})$. 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.Comment: 32 pages plus 2 pages of reference