Lie theory of finite simple groups and the generalised Roth conjecture

Details Lie theory of finite simple groups and the generalised Roth conjecture Lie theory of finite simple groups and the generalised Roth conjecture

2010-03-29

arxiv.org/abs/1003.5611v1

Mathematics

Quantum Algebra

latex 38 pages with two .pdf figures

Scientific paper

We apply a recent approach to noncommutative differential geometry on finite groups to the case of finite nonabelian simple groups and similar groups with trivial centre. The `Lie algebra' or bicovariant differential calculus here is provided by an ad-stable generating set and by analogy with Lie theory we consider when the associated Killing form is nondegenerate. We prove nondegeneracy for the case of the universal calculus whenever Roth's property holds, including for all symmetric groups $S_n$, all sporadic and most other finite simple nonabelian groups. We conjecture that nondegeneracy holds more generally and prove it for the 2-cycles calculus on any $S_n$, and by computer for all real conjugacy classes on finite simple nonabelian groups up to order 75,000. In all cases we find that the Killing form is in fact either positive definite, if the conjugacy class consists of involutions, or otherwise has zero (evenly split) signature. As an application of the Killing form we find that its eigenspaces typically decompose the conjugacy class representation into irreducibles and we explore the possibility of bijectively assigning an irreducible representation to a conjugacy class containing it. We prove that the conjugacy classes of $S_n$ containing the sign representation are those corresponding to partitions of $n$ into distinct odd parts and relate this observation to a classical identity of Euler.

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