Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions
Bibliographic record
Abstract
This is a survey paper on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. In §1, we present a detailed review (with complete proofs) of the classical structure theory of the group algebra $\mathbb{F} S_n$ of the symmetric group $S_n$ over a field $\mathbb{F}$ of characteristic 0 (or $p > n$). The goal is to obtain a constructive version of the isomorphism $\psi\colon \bigoplus_\lambda M_{d_\lambda} (\mathbb{F}) \longrightarrow \mathbb{F} S_n$ where $\lambda$ is a partition of $n$ and $d_\lambda$ counts the standard tableaux of shape $\lambda$. Young showed how to compute $\psi$; to compute its inverse, we use an efficient algorithm for representation matrices discovered by Clifton. In §2, we discuss constructive methods based on §1 which allow us to analyze the polynomial identities satisfied by a specific (non)associative algebra: fill and reduce algorithm, module generators algorithm, Bondari's algorithm for finite dimensional algebras. In §3, we study the multilinear identities satisfied by the octonion algebra $\mathbb{O}$ over a field of characteristic 0. For $n \le 6$ we compare our computational results with earlier work of Racine, Hentzel \&\ Peresi, Shestakov \&\ Zhukavets. Going one step further, we verify computationally that every identity in degree 7 is a consequence of known identities of lower degree; this result is our main original contribution. This gap (no new identities in degree 7) motivates our concluding conjecture: the known identities for $n \le 6$ generate all of the octonion identities in characteristic 0.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".