## 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)].

### MSC:

 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

### Citations:

Zbl 0923.11069; Zbl 0578.10028
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### References:

 [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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