×

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
PDF BibTeX XML Cite
Full Text: DOI

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).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.