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On Majorana representations of $$A _6$$ and $$A _7$$. (English) Zbl 1226.17023
From the summary: The Majorana representations of groups were introduced by A. A. Ivanov [The Monster group and Majorana involutions. Cambridge Tracts in Mathematics 176. Cambridge: Cambridge University Press (2009; Zbl 1205.20014)] by axiomatising some properties of the $$2A$$-axial vectors of the 196.884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by S. Sakuma [Int. Math. Res. Not. 2007, No. 9, Article ID rnm030 (2007; Zbl 1138.17013)]. This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by M. Miyamoto [J. Algebra 268, No. 2, 653–671 (2003; Zbl 1053.17018)]. Every subgroup $$G$$ of the Monster which is generated by its intersection with the conjugacy class of $$2A$$-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to $$G$$ the action of the Monster on its algebra. This representation of $$G$$ is said to be based on an embedding of $$G$$ in the Monster.
So far the Majorana representations have been classified for the groups $$G$$ isomorphic to the symmetric group $$S_4$$ of degree 4 [A. A. Ivanov et al. in J. Algebra 324, No. 9, 2432–2463 (2010; Zbl 1257.20011)], the alternating group $$A_5$$ of degree 5 [A. A. Ivanov and Á. Seress in Majorana representations of $$A_5$$, Preprint (2010)], and the general linear group $$\text{GL}_3(2)$$ in dimension 3 over the field of two elements [A. A. Ivanov and S. Shpectorov in Majorana Representations of $$L_3(2)$$, Preprint (2010)]. All these representations are based on embeddings in the Monster of either the group $$G$$ itself or of its direct product with a cyclic group of order 2.
In the present note the classification is expanded to the groups $$A_6$$ and $$A_7$$ (subject to invariance of the $$3A$$-axial vectors and the absence of $$3C$$-subalgebras).

##### MSC:
 17B69 Vertex operators; vertex operator algebras and related structures 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations
##### Keywords:
Majorana representations of groups; Monster algebra
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##### References:
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