On Majorana representations of \(A _6\) and \(A _7\).

*(English)*Zbl 1226.17023From 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).

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 |

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\textit{A. A. Ivanov}, Commun. Math. Phys. 307, No. 1, 1--16 (2011; Zbl 1226.17023)

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##### References:

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[2] | Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985) · Zbl 0568.20001 |

[3] | Ivanov, A.A.: The Monster Group and Majorana Involutions. Cambridge: Cambridge Univ. Press, 2009 · Zbl 1205.20014 |

[4] | Ivanov, A.A.: Majorana representations of A 6 involving 3C-algebras. Manuscript, 2011 · Zbl 1257.17036 |

[5] | Ivanov A.A., Pasechnik D.V., Seress Á., Shpectorov S.: Majorana representations of the symmetric group of degree 4. J. Alg. 324, 2432–2463 (2010) · Zbl 1257.20011 · doi:10.1016/j.jalgebra.2010.07.015 |

[6] | Ivanov, A.A., Shpectorov, S.: Majorana Representations of L 3(2). Preprint, 2010 · Zbl 1257.20011 |

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[8] | Miyamoto M.: Vertex operator algebras generated by two conformal vectors whose \(\tau\)-involutions generate S 3. J. Alg. 268, 653–671 (2003) · Zbl 1053.17018 · doi:10.1016/S0021-8693(03)00096-6 |

[9] | Norton, S.P.: The Monster algebra: some new formulae. In: Moonshine, the Monster and Related Topics, Contemp. Math. 193, Providence, RI: Amer. Math. Soc., 1996, pp. 297–306 · Zbl 0847.11023 |

[10] | Norton, S.P.: Anatomy of the Monster I. In: The Atlas of Finite Groups: Ten Years On, LMS Lect. Notes Ser. 249, Cambridge: Cambridge Univ. Press, 1998, pp. 198–214 · Zbl 0908.20008 |

[11] | Sakuma, S.: 6-Transposition property of \(\tau\)-involutions of Vertex Operator Algebras. International Math. Research Notes 2007, article rnm030, 19 pages, doi: 10.1093/imrn/rmn030 · Zbl 1138.17013 |

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