Modular forms and representations of the dihedral group. (English. Russian original) Zbl 0923.11070

Math. Notes 63, No. 1, 115-118 (1998); translation from Mat. Zametki 63, No. 1, 130-133 (1998).
The author continues her study of the relationship between representations of finite groups and multiplicative eta products. (Cf. the preceding review Zbl 0923.11069). Theorem: For the dihedral groups \(D_n\) with \(3\leq n\leq 23\), \(n\notin\{13, 17, 19\}\) (and only for these \(n)\), there exists a faithful representation \(\Phi\) of \(D_n\) such that, for any \(g\in D_n\), the eta product \(\eta_g(z)\) corresponding to \(\Phi(g)\) is an eigenfunction of all Hecke operators. A complete list of the functions \(\eta_g(z)\) is given. It contains all the 28 multiplicative eta products with integral weight which are known to exist [D. Dummit, H. Kisilevsky and J. McKay, Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)].


11F22 Relationship to Lie algebras and finite simple groups
11F20 Dedekind eta function, Dedekind sums
11F11 Holomorphic modular forms of integral weight
11F25 Hecke-Petersson operators, differential operators (one variable)
20C15 Ordinary representations and characters
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[1] G. V. Voskresenskaya,Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.],29, No. 2, 71–73 (1995). · Zbl 0847.11022
[2] G. V. Voskresenskaya,Mat. Zametki [Math. Notes],60, No. 2, 292–294 (1996).
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