## Abstract

In this article, we study the Ising vectors in the vertex operator algebra V^{+}∧ associated with the Leech lattice ∧. The main result is a characterization of the Ising vectors in V^{+}∧. We show that for any Ising vector e in V^{+}∧, there is a sublattice E ≅ √2E_{8} of ∧ such that e ∈ V^{+}E . Some properties about their corresponding τ -involutions in the moonshine vertex operator algebra V are also discussed.We show that there is no Ising vector of σ- type in V. Moreover, we compute the centralizer CAutV(z, τe) for any Ising vector e ∈ V^{+}∧, where z is a 2B element in AutV which fixes V^{+}∧. On the basis of this result, we also obtain an explanation for the 1A case of an observation by Glauberman-Norton (2001), which describes some mysterious relations between the centralizer of z and some 2A elements commuting z in the Monster and the Weyl groups of certain sublattices of the root lattice of type E_{8}.

Original language | English |
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Article number | rnm132 |

Journal | International Mathematics Research Notices |

Volume | 2007 |

DOIs | |

Publication status | Published - 2007 Dec 1 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)