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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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References

REFERENCES

[1] Besche, H. U., Eick, B. and O'Brien, E. The small groups library http://www.cm.tu-bs.de/ag_algebra/software/small/ Google Scholar
[2] Connes, A. Noncommutative Geometry (Academic Press, 1994).Google Scholar
[3] Conway, J. H., Curtis, R. T., Parker, R. A., Norton, S. P. and Wilson, R. A. Atlas of Finite Groups (Clarendon Press, 1985).Google Scholar
[4] Frumkin, A. Theorem about the conjugacy representation of Sn . Israel J. Math. 55 (1986), 121–128.Google Scholar
[5] Fulton, W. Young tableaux. London Mathematical Society Student Texts, no. 35 (Cambridge University Press, 1997).Google Scholar
[6] Fulton, W. and Harris, J. Representation Theory A fist Course. Graduate Texts in Mathematics 129 (Springer Verlag, 1991).Google Scholar
[7] The GAP Group GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008), http://www.gap-system.org.Google Scholar
[8] Gomez, X. and Majid, S. Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras. J. Algebra 261 (2003), 334–388.Google Scholar
[9] Heide, G. and Zalesski, A. Passman's problem on adjoint representations. In: Groups, Rings and Algebras (Proc. Conf. in Honour of D.S. Passman), Contemp. Math. 420 (2006), 163–176. Amer. Math. Soc.CrossRefGoogle Scholar
[10] Heide, G., Saxl, J., Tiep, P. and Zalesski, A. Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type. Proc. London Math. Soc. 106 (2013), 908–930.Google Scholar
[11] Leinster, T. The magnitude of metric spaces. Doc. Math. 18 (2013), 857–905.Google Scholar
[12] Majid, S. A Quantum Groups Primer. London Math. Soc. Lect. Notes no 292 (Cambridge University Press, 2002).Google Scholar
[13] Majid, S. Quantum and braided Lie-algebras. J. Geom. Phys. 13 (1994), 307–356.Google Scholar
[14] Majid, S. Solutions of the Yang–Baxter equations from braided-lie algebras and braided groups. J. Knot Theory Ramif. 4 (1995), 673–697.Google Scholar
[15] Majid, S. Noncommutative differentials and Yang–Mills on permutation groups SN . Lecture Notes Pure Appl. Math. 239 (Marcel Dekker, 2004), 189–214.Google Scholar
[16] Majid, S. Riemannian geometry of quantum groups and finite groups with nonuniversal differentials. Commun. Math. Phys. 225 (2002), 131–170.Google Scholar
[17] Majid, S. Noncommutative Riemannian geometry of graphs. J. Geom. Phys. 69 (2013), 74–93.CrossRefGoogle Scholar
[18] Majid, S. and Rietsch, K. Lie theory and coverings of finite groups. J. Algebra 389 (2013), 137–150.Google Scholar
[19] Ngakeu, F., Majid, S. and Lambert, D. Noncommutative Riemannian geometry of the alternating group A 4 . J. Geom. Phys. 42 (2002), 259–282.Google Scholar
[20] Passman, D. S. The adjoint representation of group algebras and enveloping algebras. Publ. Math. 36 (1992), 861–878.CrossRefGoogle Scholar
[21] Roth, R. L. On the conjugation representation of a finite group. Pacific J. Math. 36 (1971), 515–521.CrossRefGoogle Scholar
[22] Sagan, B. The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions, 2nd edition (Springer-Verlag, New York, 2001).CrossRefGoogle Scholar
[23] Scharf, T. Ein weiterer Beweis, dass die konjugierende Darstellung der symmetrischen Gruppen jede irreduzible Darstellung enthält. Arch. Math. 54 (1990), 427–429.Google Scholar
[24] Stein, W. A. et al. Sage Mathematics Software (Version 4.8). The Sage Development Team (2011), http://www.sagemath.org.Google Scholar
[25] Wilf, H. S. Lectures on integer partitions. Online notes of lectures given at University of Victoria, B.C. Canada (2000).Google Scholar
[26] Woronowicz, S. L. Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys. 122 (1989), 125–170.Google Scholar
[27] Zagier, D. On the distribution of the number of cycles of elements in symmetric groups. Nieuw Archief voor Wiskunde 13 (1995), 489–495.Google Scholar