2014/07/14 by Murray R. Bremner, Bremner, Murray, Sara Madariaga +3 · 1 citation
Mathematics · #16S34 #17-08 #17A30 #17A75 #17D05 #20B30 #20C40 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Primary 20C30. Secondary 16K20 #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1407.3810
openalex publication_date 2014/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In part 1, we review the structure theory of \mathbbF Sn, the group algebra of the symmetric group Sn over a field of characteristic 0. We define the images ψ(Eλij) of the matrix units Eλij (1 ≤ i, j ≤ dλ), where dλ is the number of standard tableaux of shape λ, and obtain an explicit construction of Young's isomorphism ψ\colon \bigoplusλMdλ(\mathbbF) → \mathbbF Sn. We then present Clifton's algorithm for the construction of the representation matrices Rλ(p) ∈ Mdλ(\mathbbF) for all p ∈ Sn, and obtain the reverse isomorphism ϕ\colon \mathbbF Sn → \bigoplusλMdλ(\mathbbF). In part 2, we apply the structure theory of \mathbbF Sn to the study of multilinear polynomial identities of degree n ≤ 7 for the algebra \mathbbO 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 ≤ 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 ≤ 6 generate all octonion identities in characteristic 0.