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THE MOONSHINE MODULE FOR CONWAY’S GROUP

Part of: Lie algebras and Lie superalgebras Representation theory of groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 June 2015

JOHN F. R. DUNCAN  and
SANDER MACK-CRANE
JOHN F. R. DUNCAN
Affiliation:
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA; john.duncan@case.edu, mack-crane@case.edu
SANDER MACK-CRANE
Affiliation:
Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University, Cleveland, OH 44106, USA; john.duncan@case.edu, mack-crane@case.edu

Abstract

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We exhibit an action of Conway’s group – the automorphism group of the Leech lattice – on a distinguished super vertex operator algebra, and we prove that the associated graded trace functions are normalized principal moduli, all having vanishing constant terms in their Fourier expansion. Thus we construct the natural analogue of the Frenkel–Lepowsky–Meurman moonshine module for Conway’s group. The super vertex operator algebra we consider admits a natural characterization, in direct analogy with that conjectured to hold for the moonshine module vertex operator algebra. It also admits a unique canonically twisted module, and the action of the Conway group naturally extends. We prove a special case of generalized moonshine for the Conway group, by showing that the graded trace functions arising from its action on the canonically twisted module are constant in the case of Leech lattice automorphisms with fixed points, and are principal moduli for genus-zero groups otherwise.

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11F22: Relationship to Lie algebras and finite simple groups 17B69: Vertex operators; vertex operator algebras and related structures 20C34: Representations of sporadic groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015 , e10
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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