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Standard Majorana representations of the symmetric groups. (English) Zbl 1351.05228
Let \(G\) be a finite group generated by a \(G\)-stable set \(\mathcal{T}\) of involutions. A Majorana representation of \(G\) is a quintuple \(\mathcal{R} = (G,\mathcal{T}, W, \phi,\psi)\), where \(W\) is a finite-dimensional Euclidean space equipped with a commutative (not necessarily associative) algebra product, \(\phi: G \longrightarrow \mathrm{Aut}(W)\) is a representation of \(G\) which preserves the algebra product on \(W\) and such that \(g^\phi\) is an isometry on \(W\) for each \(g\in G\), and \(\psi: \mathcal{T}\longrightarrow W\setminus\{0\}\) is an injective map such that, for every \(g\in G\) and \(t\in \mathcal{T}\), \((t^\psi)^{g^\phi} = (g^{-1}tg)^\psi\). This tuple must satisfy a certain list of conditions (see [A. Castillo-Ramirez and A. A. Ivanov, in: Groups of exceptional type, Coxeter groups and related geometries. Invited articles based on the presentations at the international conference on “Groups and geometries”, Bangalore, India, December 10–21, 2012. New Delhi: Springer. 159–188 (2014; Zbl 1341.20010)]). The fundamental example of a Majorana representation is \(\mathcal{R}_M = (M,\mathcal{T}_M,U_M,\phi_M,\psi_M)\), where \(M\) is the monster group, \(\mathcal{T}_M\) is a set of \(2A\)-involutions of \(M\), \(U_M\) is the Conway-Norton-Griess algebra, \(\phi_M\) is the faithful representation of \(M\) on \(U_M\), and \(\psi_M: \mathcal{T}_M \longrightarrow U_M\) is the map \(t\mapsto a(t)\), where \(a(t)\) is the axial vector of \(t\).
Let \(S_n\) be the symmetric group on \(n\) letters, and let \(\mathcal{T}_n\) be the set of permutations which are products of two disjoint transpositions. A Majorana representation \(\mathcal{R} = (S_n,\mathcal{T},W,\phi,\psi)\) of \(S_n\) is said to be standard if \(\mathcal{T}_n \subseteq \mathcal{T}\). In [S. P. Norton, Contemp. Math. 45, 271–285 (1985; Zbl 0577.20013)], it was shown that \(S_n\) has a standard Majorana representation if \(n \leq 12\). In the paper under review, the authors prove \(S_n\) has a standard Majorana representation if and only if \(n\leq 12\). Moreover, for \(8\leq n \leq 12\), a decomposition, in terms of the Specht modules of \(S_n\), is given for the subspace of \(W\) spanned by the set \(\mathcal{T}_n^\psi\).

MSC:
05E10 Combinatorial aspects of representation theory
05E30 Association schemes, strongly regular graphs
20C30 Representations of finite symmetric groups
20B30 Symmetric groups
17B69 Vertex operators; vertex operator algebras and related structures
05B99 Designs and configurations
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