21 involutions acting on the Moonshine module (Q1897802)

The author develops a means of constructing the Monster finite simple group \(M\) of \textit{B. Fischer} and \textit{R. Griess} [Invent. Math. 69, 1-102 (1982; Zbl 0498.20013)]. This approach provides a simpler way to describe the subgroups of \(M\) than the earlier methods. The author's point of view stems from several earlier discoveries about \(M\). First, \textit{A. A. Ivanov} [Durham Sympos., Lond. Math. Soc. Lect. Note Ser. 165, 46-62 (1992; Zbl 0821.20005)] and \textit{S. P. Norton} [ibid., 63-76 (1992; Zbl 0806.20019)] showed that the Coxeter relations of the 16-node diagram \(Y_{5,5,5}\), plus the single additional relation \((ab_1 c_1 ab_2 c_2 ab_3 c_3)^{10} = 1\) provide a presentation of the bimonster \(B = M \wr \mathbb{Z}_2\), whose existence was first conjectured by J. H. Conway. Second, \textit{J. H. Conway}, \textit{S. P. Norton} and \textit{L. H. Soicher} [Computers in algebra, Lect. Notes Pure Appl. Math. 111, 27-50 (1988; Zbl 0693.20014)] showed that \(B\) is generated by 26 involutions that correspond to 13 lines and 13 points of the projective plane of order three and that satisfy the Coxeter relations of its incidence matrix. Additionally, all symmetries of that diagram are automorphisms of \(B\). Removing the centralizer of a line \(q\) that permutes two copies of \(M\) in \(B\) yields \(M\). (Conway, Norton and Soicher used a point instead of a line.) Deletion of one line \(q\) and four points containing it from the 26 nodes leaves a diagram of nine points and twelve lines that is the diagram of the affine plane of order three. The author calls this the 21-node system. A symmetry of the 21-node system is a symmetry of the affine plane of order three. Several subgroups of \(M\) are particularly easy to describe from this point of view, which also suggests a new relation between \(M\) and the automorphism group of the Lorentzian lattice \(L\) of dimension 26. The author's construction uses 21 Leech roots (nine points and twelve lines) in \(L\) that are given by a deep-hole isotropic element of type \(E^4_6\) and that satisfy the same Coxeter relations. The author first constructs a unique orthogonal basis \(\{\alpha^p_i \mid i = 1, \dots, 24\}\) of \(C \otimes \Lambda\) (where \(\Lambda\) is the Leech lattice) up to the signs of \(\alpha^p_i\) for each point \(p\). This and the triality involution \(\sigma_p\) of \textit{I. Frenkel}, \textit{J. Lepowsky} and \textit{A. Meurman} [Vertex operator algebras and the Monster, Academic Press (1988; Zbl 0674.17001)] make it possible to obtain an automorphism of the moonshine module that does not depend on the signs of the \(\alpha^p_i\) (so is uniquely determined by each point). This also leads to a new orthogonal coordinate system for \(C \otimes \Lambda\).