Ising vectors in the vertex operator algebra V+∧ associated with the leech lattice ∧

Ching Hung Lam, Hiroki Shimakura

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13 Citations (Scopus)

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 ≅ √2E8 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 E8.

Original languageEnglish
Article numberrnm132
JournalInternational Mathematics Research Notices
Volume2007
DOIs
Publication statusPublished - 2007 Dec 1
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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