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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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##### References:
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