Abstract
We give an interpretation of the coefficients of some modular forms in terms of modular representations of symmetric groups. Using this we can obtain asymptotic formulas for the number of blocks of the symmetric group Sn over a field of characteristic p for n → ∞. For p < 7 we give simple explicit formulas for the number of blocks of defect zero. The study of the modular forms leads to interesting identities involving the Dedekind η-function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
Ch. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras, Wiley-Interscience, New York (1962).
G. de B. Robinson, Representation Theory of Symmetric Groups, Edinburgh Univ. Press (1961).
I. G. Macdonald, “Affine root systems and Dedekind's η-function,” Invent. Math.,15, 91–143 (1972).
S. Lang, Introduction to Modular Forms, Springer-Verlag, Berlin (1976).
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten, Tokyo (1971).
G. A. Lomadze, “Cusp forms of prime level and neighboring type,” Tr. Mat. Inst. Akad. Nauk GSSR,63, 36–53 (1980).
D. Kubert and S. Lang, “Units in the modular function field. II,” Math. Ann.,218, 175–189 (1975).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 74–85, 1982.
Rights and permissions
About this article
Cite this article
Klyachko, A.A. Modular forms and representations of symmetric groups. J Math Sci 26, 1879–1887 (1984). https://doi.org/10.1007/BF01670574
Issue Date:
DOI: https://doi.org/10.1007/BF01670574