On Majorana Representations of A6 and A7

• Published in: Communications in mathematical physics Vol. 307; no. 1; pp. 1 - 16
• Main Author: Ivanov, A. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2011; Springer
• Subjects: Classical and Quantum Gravitation; Complex Systems; Exact sciences and technology; Functional analysis; Group theory; Group theory and generalizations; Mathematical analysis; Mathematical and Computational Physics; Mathematical methods in physics; Mathematical Physics; Mathematics; Operator theory; Other topics in mathematical methods in physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Sciences and techniques of general use; Theoretical; Dihedral Group; Conjugacy Class; Vertex Operator Algebra; Fusion Rule; Linear Span; Embedding; Ordered group; Cyclic group; Mathematical physics; Classification
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-011-1298-6

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009 ) by axiomatising some properties of the 2 A -axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007 ). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003 ). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2 A -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 (Ivanov et al. in J Alg 324:2432–2463, 2010 ), the alternating group A 5 of degree 5 (Ivanov AA, Seress Á in Majorana Representations of A 5 , 2010 ), and the general linear group GL 3 (2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L 3 (2), 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. The dimensions and shapes of these representations are given in the following table: “What is our life? A game!” (A.S. Pushkin, “The Queen of Spades”) Shape (2 A , 3 A ) (2 A , 3 C ) (2 B , 3 A ) (2 B , 3 C ) S 4 13 9 13 6 A 5 26 20 46 21 L 3 (2) 49 21 – – In the present note the classification is expanded to the groups A 6 and A 7 (subject to invariance of the 3 A -axial vectors and the absence of 3 C -subalgebras).