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Lie theory of finite simple groups and the Roth property

Published online by Cambridge University Press:  09 January 2017

J. LÓPEZ PEÑA ,
S. MAJID  and
K. RIETSCH
J. LÓPEZ PEÑA
Affiliation:
University College London, Department of Mathematics, Gower Street, London WC1E 6BT. e-mail: [email protected]
S. MAJID
Affiliation:
Queen Mary University of London, School of Mathematical Sciences, Mile End Rd, London E1 4NS. e-mail: [email protected]
K. RIETSCH
Affiliation:
Kings College London, Department of Mathematics, The Strand, London WC2R 2LS. e-mail: [email protected]
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Abstract

In noncommutative geometry a ‘Lie algebra’ or bidirectional bicovariant differential calculus on a finite group is provided by a choice of an ad-stable generating subset $\mathcal{C}$ stable under inversion. We study the associated Killing form K. For the universal calculus associated to $\mathcal{C}$ = G \ {e} we show that the magnitude $\mu=\sum_{a,b\in\mathcal{C}}(K^{-1})_{a,b}$ of the Killing form is defined for all finite groups (even when K is not invertible) and that a finite group is Roth, meaning its conjugation representation contains every irreducible, iff μ ≠ 1/(N − 1) where N is the number of conjugacy classes. We show further that the Killing form is invertible in the Roth case, and that the Killing form restricted to the (N − 1)-dimensional subspace of invariant vectors is invertible iff the finite group is an almost-Roth group (meaning its conjugation representation has at most one missing irreducible). It is known [9, 10] that most nonabelian finite simple groups are Roth and that all are almost Roth. At the other extreme from the universal calculus we prove that the 2-cycles conjugacy class in any Sn has invertible Killing form, and the same for the generating conjugacy classes in the case of the dihedral groups D2n with n odd. We verify invertibility of the Killing forms of all real conjugacy classes in all nonabelian finite simple groups to order 75,000, by computer, and we conjecture this to extend to all nonabelian finite simple groups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 301 - 340
Copyright
Copyright © Cambridge Philosophical Society 2017 

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