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Structure theory for the group algebra of the symmetric group, with applications to polynomial identities for the octonions. (English) Zbl 1438.17001

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.

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