Ramifications of Ramanujan's work on \(\eta\)-products
type: Item
instance of: scholarly article
title: Ramifications of Ramanujan's work on \(\eta\)-products
zbMATH Open document ID: 0693.10022
DOI: 10.1007/BF02864394
author: John McKay
author: David J. Ford
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1989-01-01T00:00:00Z
review text: This short note summarizes some observations and calculations relating certain of the sporadic groups with modular forms whose Fourier coefficients are multiplicative (eigenfunctions of Hecke operators). The principal result is based on the fact that the shape of a permutation (the partition of its degree formed by its disjoint cycle lengths) can be used to construct a modular form defined as a product of Dedekind \(\eta\)- functions: for x a permutation, \[ shape(x)=(1^{e_ 1},2^{e_ 2},...,r^{e_ r})\quad \to \quad f_ x(z)=\sum_{t\geq 1}\eta (tz)^{e_ t}. \] It turns out that for all the permutations x in the Mathieu group \(M_{24}\) of degree 24, \(f_ x(z)\) has multiplicative coefficients. Similar observations are made concerning the monster group \(F_ 1\) and further speculation is given concerning the connection of these phenomena with conformal quantum field theory.
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 20C30
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 4134146
zbMATH Keywords: multiplicative Fourier coefficients
zbMATH Keywords: finite simple groups
zbMATH Keywords: shape of permutation
zbMATH Keywords: sporadic groups
zbMATH Keywords: modular forms
zbMATH Keywords: Dedekind \(\eta\)-functions
zbMATH Keywords: Mathieu group
zbMATH Keywords: monster group \(F_ 1\)
zbMATH Keywords: conformal quantum field
Wikidata QID: Q29308470
reviewed by: James Lee Hafner
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4134146