On McKay's connection between the affine \(E_8\) diagram and the Monster (Q2782385)

Let \(\widetilde E_8\) be the affine \(E_8\)-diagram, \(\alpha_1,\dots,\alpha_8\) be fundamental roots in a root system for \(E_8\) and \(\alpha_M\) be the maximal root for the system. Then \(\widetilde E_8\) is obtained by adding a node for \(\alpha_0=-\alpha_M\) to the \(E_8\)-diagram and \(\sum_{i=0}^8c_i\alpha_i=0\) for some positive integers \(c_i\), \(c_0=1\).NEWLINENEWLINENEWLINEMcKay has discovered mysteries about \(\widetilde E_8\) and the Monster sporadic group \(\mathbf M\). Let \(t\) be a Fischer involution in \(\mathbf M\) so that \(C_t=C_{\mathbf M}(t)\simeq 2\cdot B\). Then \(t^{\mathbf M}\) splits into 9 suborbits under \(C_t\), say \(t^{\mathbf M}=t_0^{C_t}\cup t_1^{C_t}\cup\cdots\cup t_8^{C_t}\). For some renumbering of \(t_0,\dots,t_8\) the following hold:NEWLINENEWLINENEWLINE(A) the order of the product \(tt_i\) is \(c_i\) for \(i=0,\dots,8\);NEWLINENEWLINENEWLINE(B) for \(i\neq j\), \(tt_i\) is not conjugate to \(tt_j\) under \(\mathbf M\).NEWLINENEWLINENEWLINEThe analogue of (A) is true for the affine \(E_6\) diagram. Let \(u=tt_2\), \(H=C_{\mathbf M}(u)\langle t \rangle\), \(C=C_{\mathbf M}(t)\cap C_{\mathbf M}(u)\). Then \(u\) has type 3A in the ATLAS notation and \(t^H\) splits into 8 suborbits with representatives \(t_0',\dots,t_7'\) such that:NEWLINENEWLINENEWLINE(A') \(c_i\) is the order of the coset of \(tt_i'\) in \(H/\langle u \rangle\).NEWLINENEWLINENEWLINEThe analogue of (B) is false.NEWLINENEWLINENEWLINEThe action of \(u\) induces an automorphism of order 3 of the \(\widetilde E_6\) diagram and we obtain a ``folded'' \(\widetilde G_2\) diagram. The factor group \(\overline H=H/\langle u\rangle\) is isomorphic to \(Fi_{24}\) and contains the involution \(\overline t=t\langle u\rangle\). The analogues of (A) and (B) are valid for \(C_{\overline H}(\overline t)\) with respect to the \(\widetilde G_2\) diagram.NEWLINENEWLINENEWLINEIn the case \(u=tt_1\) the suborbits of \(H=C_{\mathbf M}(u)\) give the affine \(\widetilde E_7\) diagram, which folds under a symmetry of order 2 to the \(\widetilde F_4\) diagram. The analogues of (A) and (B) are valid for \(H/\langle u \rangle\simeq B\).NEWLINENEWLINENEWLINEIn this paper are obtained additional connections between the products \(tt_i\) and \(\widetilde E_8\) for \(i>2\). It is discovered that the \(tt_i\) cases for \(i=3,4,5\) satisfy analogues of all the results above, while the other cases satisfy partial analogues of the ``folding'' results.NEWLINENEWLINENEWLINEIn any case the group \(K=C_{\mathbf M}(\langle u,t\rangle)\) contains an element \(z\) of type 2B in \(\mathbf M\) such that \(C_K(z)\) has a section which is isomorphic to ``half'' the Weyl group of a quotient for the affine \(E_8\)-diagram.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].