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 $\psi (E_{ij}^{\lambda })$ of the matrix units $E_{ij}^{\lambda }$ ($1\le i,j\le d_{\lambda }$), where $d_{\lambda }$ is the number of standard tableaux of shape $\lambda$, and obtain an explicit construction of Young's isomorphism $\psi :{⨁}_{\lambda }{M}_{d_{\lambda }}(\mathbb{F})\to \mathbb{F}{S}_n$. 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 $\varphi :\mathbb{F}{S}_n\to {⨁}_{\lambda }{M}_{d_{\lambda }}(\mathbb{F})$. 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.

32 pages plus 2 pages of references

Cited by in corpus (5)

• Classification of regular parametrized one-relation operads
• Lie and Jordan products in interchange algebras
• Higher polynomial identities for mutations of associative algebras
• Special Identities for Comtrans Algebras
• Splitting of operations for alternative and Malcev structures