Majorana Theory

Alexander A. Ivanov (Imperial College London, UK)

The Monster group $M$, which is the largest among the 26 sporadic simple groups is the automorphism group of 196 884-dimensional Conway--Griess--Norton algebra (simply called the Monster algebra). There is a remarkable correspondance between the so-called $2A$-involutions in $M$ and certain idempotents in the Monster algebra (we refer to these idempotents as Majorana axes). The isomorphism types of the subalgebras in the Monster algebra generated by pairs of Majorana axes were calculated by S. Norton a while ago (there are precisely nine isomorphism types). More recently these nine algebras were characterized by S. Sakuma in the context of Vertex Operator Algebras, relying on earlier work by M. Miyamoto. The properties of Monster algebras used in the proof of Sakuma's theorem are rather elementary and they have been axiomatized under the name of Majorana representations. In this terminology Sakuma's theorem amounts to classification of the Majorana representations of the dihedral groups together with a remark that all the representations are based on embeddings into the Monster. In the present paper it is shown that the alternating group $A_5$ of degree 5 possesses precisely two Majorana representations, both based on embeddings into the Monster. The dimensions of the representations are 20 and 26; the scalar squares of the identities are $10$ and $\frac{72}{7}$, respectively (in the Vertex Operator Algebra context these numbers are doubled central charges).