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Modular forms and regular representations of groups of order 24. (English. Russian original) Zbl 0923.11069

Math. Notes 60, No. 2, 216-218 (1996); translation from Mat. Zametki 60, No. 2, 292-294 (1996).
Theorem: Let \(G\) be a grup of order 24, let \(\Phi\) be its regular representation, and let \(P_g(X)= \prod_k(X^{a_k} -1)^{t_k}\) be the characteristic polynomial of \(\Phi(g)\) for \(g\in G\); then the eta product \(\eta_g(z)=\prod_k(\eta(a_kz))^{t_k}\) is an eigenfunction of all Hecke operators. It is known [D. Dummit, H. Kisilevsky and J. McKay, Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)] that there are exactly 30 eta products \(\prod_j \eta(n_jz)\) with \(n_j\in\mathbb{N}\), \(\sum_jn_j=24\), which have multiplicative coefficients. Only 8 of them appear in the form \(\eta_g (z)\) for groups of order 24; a complete list is given. In an earlier note [Funct. Anal. Appl. 29, No. 2, 129-130 (1995); translation from Funkts. Anal. Prilozh. 29, No. 2, 71-73 (1995; Zbl 0847.11022)] the author treated the same problem for finite subgroups of \(SL(5,\mathbb{C})\).

MSC:

11F22 Relationship to Lie algebras and finite simple groups
11F20 Dedekind eta function, Dedekind sums
11F25 Hecke-Petersson operators, differential operators (one variable)
20C15 Ordinary representations and characters
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References:

[1] G. Mason,Math. Ann.,283, 381–409 (1989). · Zbl 0636.10021
[2] D. Dummit, H. Kisilevsky and J. McKay,Contemp. Math.,45, 89–98 (1985).
[3] G. V. Voskresenskaya, ”Finite subgroups of SL(5,\(\mathbb{C}\)) and modular forms,”Dep. VINITI, No. 610-B93, Moscow, 1993.
[4] G. Mason,Contemp. Math.,45, 223–244 (1985).
[5] G. V. Voskresenskaya,Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],29, No. 2, 71–73 (1995). · Zbl 0847.11022
[6] H. S. Coxeter and W. O. Moser,Generators and Relations for Discrete Groups, Springer-Verlag, New York (1972). · Zbl 0239.20040
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