The Leech lattice $\Lambda$ and the Conway group $\cdot \mathrm {O}$ revisited
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- by John N. Bray and Robert T. Curtis PDF
- Trans. Amer. Math. Soc. 362 (2010), 1351-1369
Abstract:
We give a new, concise definition of the Conway group $\cdot \mathrm {O}$ as follows. The Mathieu group $\mathrm {M}_{24}$ acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of $\left ({24}\atop {4} \right )$ tetrads. We use this action to define a progenitor $P$ of shape $2^{\star \left ( 24 \atop 4 \right )}:\mathrm {M}_{24}$; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring $P$ by this relator results in $\cdot \mathrm {O}$. Consideration of the lowest dimension in which $\cdot \mathrm {O}$ can act faithfully produces Conway’s elements $\xi _T$ and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under $\cdot \mathrm {O}$ of the integral vectors in ${\mathbb R}_{24}$.References
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Additional Information
- John N. Bray
- Affiliation: School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, United Kingdom
- Email: j.n.bray@qmul.ac.uk
- Robert T. Curtis
- Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom
- MR Author ID: 198799
- Email: r.t.curtis@bham.ac.uk
- Received by editor(s): November 23, 2007
- Received by editor(s) in revised form: January 7, 2008
- Published electronically: October 20, 2009
- © Copyright 2009 by the authors
- Journal: Trans. Amer. Math. Soc. 362 (2010), 1351-1369
- MSC (2000): Primary 20D08; Secondary 20F05
- DOI: https://doi.org/10.1090/S0002-9947-09-04726-6
- MathSciNet review: 2563732
Dedicated: Dedicated to John Horton Conway as he approaches his seventieth birthday.