Majorana representations of the symmetric group of degree 4

The Monster group M   acts on a real vector space VMVM of dimension 196,884 which is the sum of a trivial 1-dimensional module and a minimal faithful M-module. There is an M  -invariant scalar product (,) on VMVM, an M  -invariant bilinear commutative non-associative algebra product ⋅ on VMVM (commonly known as the Conway–Griess–Norton algebra), and a subset A   of VM∖{0}VM∖{0} indexed by the 2A-involutions in M. Certain properties of the quintetM=(M,VM,A,(,),⋅) have been axiomatized in Chapter 8 of Ivanov (2009) [Iv09] under the name of Majorana representation of M. The axiomatization enables one to study Majorana representations of an arbitrary group G (generated by its involutions). A representation might or might not exist, but it always exists whenever G is a subgroup in M generated by the 2A-involutions contained in G. We say that thus obtained representation is based on an embedding of G in the Monster. The essential motivation for introducing the Majorana terminology was the most remarkable result by S. Sakuma (2007) [Sak07] which gave a classification of the Majorana representations of the dihedral groups. There are nine such representations and every single one is based on an embedding in the Monster of the relevant dihedral group. It is a fundamental property of the Monster that its 2A-involutions form a class of 6-transpositions and that there are precisely nine M-orbits on the pairs of 2A-involutions (and also on the set of 2A-generated dihedral subgroups in M  ). In the present paper we are making a further step in building up the Majorana theory by classifying the Majorana representations of the symmetric group S4S4 of degree 4. We prove that S4S4 possesses precisely four Majorana representations. The Monster is known to contain four classes of 2A  -generated S4S4-subgroups, so each of the four representations is based on an embedding of S4S4 in the Monster. The classification of 2A  -generated S4S4-subgroups in the Monster relies on calculations with the character table of the Monster. Our elementary treatment shows that there are (at most) four isomorphism types of subalgebras in the Conway–Griess-Norton algebra of the Monster generated by six Majorana axial vectors canonically indexed by the transpositions of S4S4. Two of these subalgebras are 13-dimensional, the other two have dimensions 9 and 6. These dimensions, not to mention the isomorphism type of the subalgebras, were not known before.