Abstract
Completely replicable functions play an important role in number theory and finite group theory, in particular the Monstrous Moonshine. In this paper, we give a characterization of completely replicable functions by certain symmetries.
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Alexander, D., Cummins, C., McKay, J., Simons, C.: Completely replicable functions. In: Groups, Combinatorics and Geometry, London Mathematical Society Lecture Note Series, vol. 165, pp. 87–95 (1992)
Borcherds, R.E.: Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405–444 (1992)
Carnahan, S.: Generalized moonshine I: genus-zero functions. Algebra Number Theory 4, 649–679 (2010)
Conway, J.H., Norton, S.P.: Monstrous moonshine. Bull. Lond. Math. Soc. 11, 308–339 (1979)
Cummins, C.J.: Some comments on replicable functions. In: Modern Trends in Lie Algebra Representation Theory, pp. 48–55. Kingston, ON (1993)
Cummins, C.J., Gannon, T.: Modular equations and the genus zero property of moonshine functions. Invent. Math. 129(3), 413–443 (1997)
Cummins, C.J., Norton, S.P.: Rational Hauptmoduls are replicable. Can. J. Math. 47(6), 1201–1218 (1995)
Faber, G.: Über Tschebycheffsche Polynome. Crelle 150, 79–106 (1920)
Heim, B., Kaiser, C., Murase, A.: Modular curves and symmetries of Hecke type. Int. J. Math. 29(07), 1850045 (2018)
Kozlov, D.N.: On Completely Replicable Functions and Extremal Poset Theory. M. Sc. Thesis, Department of Mathematics, University of Lund, Sweden (1994)
Kozlov, D.N.: On functions satisfying modular equations for infinitely many primes. Can. J. Math. 51, 1020–1034 (1999)
Martin, Y.: On modular invariance of completely replicable functions. Contemp. Math. 193, 263–286 (1996)
Norton, S.P.: More on Moonshine, Computational Group Theory (Durham, 1982), pp. 185–193. Academic Press, London (1984)
Schiffer, M.: Faber polynomials in the theory of univalent functions. Bull. AMS 54, 503–517 (1948)
Acknowledgements
The authors would like to thank the referee for carefully reading our manuscript and for giving valuable suggestions. The first author thanks the Max Planck Institute for Mathematics for support and an invitation in July and August 2017. He also thanks Kyoto Sangyo University and Professor Murase for an invitation in January 2018 to extend our collaboration. He also thanks Prof. Krieg for useful discussions at the Graduate school: Experimental and constructive algebra, at the RWTH Aachen in 2018 and the German University of Technology in Oman for work leave. The second author is partially supported by Grants-in-Aids from JSPS (17K05186).
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Communicated by Jens Funke.
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Heim, B., Murase, A. Completely replicable functions and symmetries. Abh. Math. Semin. Univ. Hambg. 89, 169–177 (2019). https://doi.org/10.1007/s12188-019-00212-9
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DOI: https://doi.org/10.1007/s12188-019-00212-9