McKay's \(E_6\) observation on the largest Fischer group
type: Item
instance of: scholarly article
title: McKay's \(E_6\) observation on the largest Fischer group
zbMATH Open document ID: 1258.17023
author: Gerald Höhn
author: Ching Hung Lam
author: Hiroshi Yamauchi
published in: Communications in Mathematical Physics
publication date: 2012-03-23T00:00:00Z
review text: 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 \(…
reviewed by: Hiromichi Yamada
Mathematics Subject Classification ID: 17B69
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 6018047
zbMATH Keywords: vertex operator algebra
zbMATH Keywords: Virasoro vector
zbMATH Keywords: Fischer group
MaRDI profile type: Publication
full work available at URL: s00220-011-1413-8
OpenAlex ID: W2043629883
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DOI: 10.1007/S00220-011-1413-8
description: scientific article; zbMATH DE number 6018047