McKay's \(E_6\) observation on the largest Fischer group
A mysterious correspondence between the conjugacy classes of products of pairs of involutions in the Monster \(\mathbb{M}\) (resp. the Baby Monster \(\mathbb{B}\), the largest Fischer \(3\)-transposition group \(\mathrm{Fi}_{24}\)) and the nodes of the affine \(E_8\) (resp. \(E_7\), \(E_6\)) diagram is called McKay's observation. The vertex operator algebra theoretical approach to the observation for the Monster case was initiated by \textit{C.H. Lam, H. Yamada} and \textit{H. Yamauchi} [Trans. Am. Math. Soc. 359, No. 9, 4107--4123 (2007; Zbl 1139.17011), IMRP, Int. Math. Res. Pap. 2005, No. 3, 117--181 (2005; Zbl 1082.17015)]. The argument is based on a one-to-one correspondence between the \(2A\)-elements of the Monster and the simple \(c = 1/2\) Virasoro vectors in the moonshine vertex operator algebra \(V^\natural\). Here a simple \(c = 1/2\) Virasoro vector \(e\) means that the vertex operator subalgebra \(\mathrm{Vir}(e)\) generated by \(e\) is a simple Virasoro vertex operator algebra \(L(1/2,0)\) of central charge \(c = 1/2\). The correspondence is given by \(\tau_e \leftrightarrow e\), where \(\tau_e\) is the Miyamoto \(\tau\)-involution associated to \(e\). The main ingredient is a detailed analysis of the vertex operator subalgebra \(U_{nX}\) defined as the commutant subalgebra in the vertex operator algebra \(V_{\sqrt{2}E_8}\) associated to \(\sqrt{2}\) times the \(E_8\) lattice for each node \(nX\) of the affine \(E_8\) diagram in McKay's observation. The case for the Baby Monster \(\mathbb{B}\) was studied in the authors' previous paper [Int. Math. Res. Not. 23, 166--212 (2012; Zbl 1267.17033)], where the commutant subalgebra \(V\mathbb{B}^\natural\) of \(\mathrm{Vir}(e)\) in \(V^\natural\) is considered. The automorphism group of \(V\mathbb{B}^\natural\) is isomorphic to the Baby Monster. Note that the centralizer \(C_{\mathbb{M}}(\tau_e)\) of \(\tau_e\) in \(\mathbb{M}\) is a \(2\)-fold cover \(2.\mathbb{B}\) of \(\mathbb{B}\). Using the embedding \(\mathrm{Vir}(e) \otimes V\mathbb{B}^\natural \hookrightarrow V^\natural\) and the structure of \(U_{2A}\), the authors establish a one-to-one correspondence between the \(2A\)-elements of the Baby Monster and the simple \(c=7/10\) Virasoro vectors of \(\sigma\)-type in \(V\mathbb{B}^\natural\) and prove McKay's observation for \(\mathbb{B}\). In the paper under review the authors investigate the case for \(\mathrm{Fi}_{24}\). Let \(g\) be a \(3A\)-element of the Monster. Then the normalizer \(N_{\mathbb{M}}(\langle g \rangle)\) of the cyclic group \(\langle g \rangle\) generated by \(g\) is a \(3\)-fold cover \(3.\mathrm{Fi}_{24}\) of \(\mathrm{Fi}_{24}\). It is shown that the centralizer \(C_{\mathbb{M}}(g)\) fixes a unique \(c=4/5\) extended Virasoro vertex operator algebra \(\mathcal{W}(4/5) = L(4/5,0) \oplus L(4/5,3)\). Let \(VF^\natural\) be the commutant subalgebra of \(\mathcal{W}(4/5)\) in \(V^\natural\), on which \(\mathrm{Fi}_{24}\) acts. Also recall that \(U_{3A}\) contains \(\mathcal{W}(4/5) \otimes \mathcal{W}(6/7)\), where \(\mathcal{W}(6/7) = L(6/7,0) \oplus L(6/7,5)\). Main results of the paper are the following: (i) There is a one-to-one correspondence between the \(2C\)-elements of \(\mathrm{Fi}_{24}\) and the simple \(c = 6/7\) Virasoro vectors of \(\sigma\)-type in \(VF^\natural\) given by \(\sigma_u \leftrightarrow u\), where \(\sigma_u\) is the Miyamoto \(\sigma\)-involution associated to \(u\). (ii) The vertex operator subalgebra \(U_{F(nX)}\) defined as the commutant subalgebra in the vertex operator algebra \(V_{\sqrt{2}E_6}\) for each node \(nX = 1A, 2A, 3A\) of the affine \(E_6\) diagram can be embedded in \(VF^\natural\). (iii) The product \(\sigma_u \sigma_{u'}\) of two \(\sigma\)-involutions \(\sigma_u\) and \(\sigma_{u'}\) belongs to the conjugacy class \(nX\) of \(\mathrm{Fi}_{24}\), where \(u\) and \(u'\) are a certain pair of simple \(c = 6/7\) Virasoro vectors of \(\sigma\)-type in \(U_{F(nX)}\).
scientific article; zbMATH DE number 6018047
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