Bremner, Murray R.; Madariaga, Sara; Peresi, Luiz A. Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions. (English) Zbl 1438.17001 Commentat. Math. Univ. Carol. 57, No. 4, 413-452 (2016). In 1950, A. I. Mal’tsev [Mat. Sb., N. Ser. 26(68), 19–33 (1950; Zbl 0039.26601)] and W. Specht [Math. Z. 52, 557–589 (1950; Zbl 0032.38901)] independently discovered that the description of the polynomial identities of an algebra over a field of characteristic 0 is equivalent to the description of the multilinear identities. They suggested to apply the representation theory of the symmetric group \(S_n\) to the study of the polynomial identities. Later, the method of \(S_n\)-representations was further developed by A. Regev, starting with his proof that the tensor product of associative PI-algebras is PI-again [Bull. Am. Math. Soc. 77, 1067–1069 (1971; Zbl 0225.16012); Isr. J. Math. 11, 131–152 (1972; Zbl 0249.16007)]. In the late 1970s, the computational implementation of representation theory of \(S_n\) for the purposes of polynomial identities of (nonassociative) algebras was initiated by I. R. Hentzel [Ring theory. Proc. Ohio Univ. Conf., Athens 1976, Lect. Notes Pure Appl. Math. 25, 133–141 (1977; Zbl 0356.17002); “Processing identities by group representation”, in: R. E. Beck and B. Kolman (eds.), Computers in nonassociative rings and algebras. New York: Academic Press, 13–40 (1977; Zbl 0372.17001)].The paper under review is a survey on applications of the representation theory of the symmetric group to the theory of polynomial identities for associative and nonassociative algebras. The paper starts with a detailed review (with complete proofs) of the classical structure theory of the group algebra \(\mathbb{F}S_n\) over a field \(\mathbb{F}\) of characteristic 0 (or \(p>n\)). The goal is to obtain a constructive version of the isomorphism \[ \psi:\bigoplus_{\lambda}M_{d_{\lambda}}(\mathbb{F})\to\mathbb{F}S_n, \] where \(\lambda\) is a partition of \(n\) and \(d_{\lambda}\) is the number of standard tableaux of shape \(\lambda\). The way how to compute \(\psi\) goes back to the work of Young. For the computation of its inverse, the authors describe the efficient algorithm for representation matrices discovered by J. M. Clifton in his Ph.D. thesis, see [Proc. Am. Math. Soc. 83, 248–250 (1981; Zbl 0443.20013)]. Further, the authors discuss constructive methods which allow to analyze the polynomial identities satisfied by a specific (non)associative algebra: fill and reduce algorithm, module generators algorithm, and the algorithm from the Ph.D. thesis of S. Bondari for finite dimensional algebras, see [Linear Algebra Appl. 258, 233–249 (1997; Zbl 0884.15009)]. Finally, they study the multilinear identities satisfied by the octonion algebra \(\mathbb{O}\) over a field of characteristic 0. For \(n\leq 6\), the computational results described in the paper are compared with earlier work of M. L. Racine [J. Algebra 115, No. 1, 251–260 (1988; Zbl 0651.17012)], I. R. Hentzel and L. A. Peresi [J. Algebra 188, No. 1, 292–309 (1997; Zbl 0890.17001)], I. P. Shestakov and N. Zhukavets [J. Pure Appl. Algebra 213, No. 4, 479–492 (2009; Zbl 1241.17033)]. The authors go one step further, and verify computationally that every identity in degree 7 is a consequence of known identities of lower degree; this result is their main original contribution. This gap (no new identities in degree 7) motivates the concluding conjecture: the known identities for \(n\leq 6\) generate all of the octonion identities in characteristic 0. Reviewer: Vesselin Drensky (Sofia) Cited in 7 Documents MSC: 17-04 Software, source code, etc. for problems pertaining to nonassociative rings and algebras 20C30 Representations of finite symmetric groups 16R10 \(T\)-ideals, identities, varieties of associative rings and algebras 16S34 Group rings 16Z05 Computational aspects of associative rings (general theory) 17-08 Computational methods for problems pertaining to nonassociative rings and algebras 17A50 Free nonassociative algebras 17A75 Composition algebras 17B01 Identities, free Lie (super)algebras 17C05 Identities and free Jordan structures 17D05 Alternative rings 68W30 Symbolic computation and algebraic computation Keywords:symmetric group; group algebra; Young diagrams; standard tableaux; idempotents; matrix units; two-sided ideals; Wedderburn decomposition; representation theory; Clifton’s algorithm; computer algebra; polynomial identities; nonassociative algebra; octonions Citations:Zbl 0039.26601; Zbl 0032.38901; Zbl 0225.16012; Zbl 0249.16007; Zbl 0356.17002; Zbl 0372.17001; Zbl 0443.20013; Zbl 0884.15009; Zbl 0651.17012; Zbl 0890.17001; Zbl 1241.17033 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).