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Majorana representations of the symmetric group of degree 4. (English) Zbl 1257.20011
Summary: The Monster group \(M\) acts on a real vector space \(V_M\) 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 \(V_M\), an \(M\)-invariant bilinear commutative non-associative algebra product \(\cdot\) on \(V_M\) (commonly known as the Conway-Griess-Norton algebra), and a subset \(A\) of \(V_M\setminus\{0\}\) indexed by the \(2A\)-involutions in \(M\). Certain properties of the quintet \(\mathcal M=(M,V_M,A,(\,,\,),\cdot)\) have been axiomatized in Chapter 8 of A. A. Ivanov, The Monster group and Majorana involutions [Cambridge Tracts in Mathematics 176. Cambridge: Cambridge University Press (2009; Zbl 1205.20014)] 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 [Int. Math. Res. Not. 2007, No. 9, Article ID rnm030 (2007; Zbl 1138.17013)] 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 \(S_4\) of degree 4. We prove that \(S_4\) possesses precisely four Majorana representations. The Monster is known to contain four classes of \(2A\)-generated \(S_4\)-subgroups, so each of the four representations is based on an embedding of \(S_4\) in the Monster. The classification of \(2A\)-generated \(S_4\)-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 \(S_4\). 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.

20C34 Representations of sporadic groups
20C30 Representations of finite symmetric groups
20D08 Simple groups: sporadic groups
17B69 Vertex operators; vertex operator algebras and related structures
Full Text: DOI
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