Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2014
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| _version_ | 1866913948860678144 |
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| author | Bremner, Murray Madariaga, Sara Peresi, Luiz A. |
| author_facet | Bremner, Murray Madariaga, Sara Peresi, Luiz A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1407_3810 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions Bremner, Murray Madariaga, Sara Peresi, Luiz A. Rings and Algebras Representation Theory Primary 20C30. Secondary 16K20, 16S34, 17A30, 17A75, 17D05, 17-08, 20B30, 20C40 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. |
| title | Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions |
| topic | Rings and Algebras Representation Theory Primary 20C30. Secondary 16K20, 16S34, 17A30, 17A75, 17D05, 17-08, 20B30, 20C40 |
| url | https://arxiv.org/abs/1407.3810 |