``Moonshine'' of finite groups.
type: Item
instance of: scholarly article
title: ``Moonshine'' of finite groups.
zbMATH Open document ID: 1218.20001
DOI: 10.4171/090
author: Koichiro Harada
published in: EMS Series of Lectures in Mathematics
publication date: 2010-10-05T00:00:00Z
review text: This book is based on the author's lecture notes for a two-quarter course at Ohio State University in 1983-84. The term ``moonshine'' means a mysterious relation between finite groups and modular functions. The moonshine for the Monster simple group, which is the original one, was studied by \textit{J. H. Conway} and \textit{S. P. Norton} [Bull. Lond. Math. Soc. 11, 308-339 (1979; Zbl 0424.20010)]. Since then, a number of works have been done on the subject, see for example \textit{I. Frenkel, J. Lepowsky} and \textit{A. Meurman}, [Vertex operator algebras and the Monster. Pure and Applied Mathematics 134. Boston: Academic Press (1988; Zbl 0674.17001)] and \textit{T. Gannon}, [Moonshine beyond the Monster. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press (2006; Zbl 1146.11026)]. Although the lecture notes were written more than 25 years ago, the contents of the present book are essentially unchanged and only the bibliography is updated. This is mainly because the author discusses very basic materials. The book consists of four chapters. The first two are preparatory ones and proofs for theorems mentioned there are usually omitted. Chapter 1, entitled ``Modular functions and modular forms,'' is devoted to standard descriptions of modular functions and modular forms. Chapter 2, entitled ``Dedekind eta function,'' deals with basic properties of the Dedekind eta function \(\eta(z)\). In Chapter 3, entitled ``Moonshine of finite groups,'' the author discusses the main ingredients including generalized partitions and frame shapes. Chapter 4, entitled ``Multiplicative product of eta function,'' reviews a conjecture of J. McKay on products of \(\eta(z)\). The author follows \textit{D. Dummit, H. Kisilevsky} and \textit{J. McKay} [Finite groups -- coming of age, Contemp. Math. 45, 89-98 (1985; Zbl 0578.10028)] and \textit{M. Koike} [Nagoya Math. J. 95, 85-89 (1984; Zbl 0548.10018)] for its proof. The book serves as a gentle introduction to …
Mathematics Subject Classification ID: 20-02
Mathematics Subject Classification ID: 11F22
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 17B69
zbMATH DE Number: 5796327
zbMATH Keywords: moonshine
zbMATH Keywords: modular functions
zbMATH Keywords: modular forms
zbMATH Keywords: Monster simple group
zbMATH Keywords: eta-function
reviewed by: Hiromichi Yamada
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/090
OpenAlex ID: W1582206837
description: scientific article; zbMATH DE number 5796327