"Moonshine" of finite groups / Koichiro Harada.

Format:

Book

Author/Creator:

Harada, Koichiro, 1941-

Contributor:

Rosengarten Family Fund.

Series:

EMS series of lectures in mathematics

Language:

English

Subjects (All):

Finite groups.

Modular functions.

Vertex operator algebras.

Mathematical physics.

Group theory.

Physical Description:

vi, 76 pages : illustrations ; 24 cm.

Place of Publication:

Zürich : European Mathematical Society, ©2010.

Summary:

This is an almost verbatim reproduction of the author's lecture notes written in 1983-84 at the Ohio State University, Columbus, Ohio, USA. A substantial update is given in the bibliography. Over the last 20 plus years, there has been an energetic activity in the field of finite simple group theory related to the monster simple group. Most notably, influential works have been produced in the theory of vertex operator algebras whose research was stimulated by the moonshine of the finite groups. Still, we can ask the same questions now just as we did some 30-40 years ago: What is the monster simple group? Is it really related to the theory of the universe as it was vaguely so envisioned? What lays behind the moonshine phenomena of the monster group? It may appear that we have only scratched the surface. These notes are primarily reproduced for the benefit of young readers who wish to start learning about modular functions used in moonshine.

Contents:

Modular functions and modular forms

Dedekind eta function

"Moonshine" of finite groups

Multiplicative product of n functions

Appendix. Genus zero discrete groups.

1 Modular functions and modular forms

1.1 Linear fractional transformations

1.2 Fundamental domains, invariant measures

1.3 Riemann surfaces associated with Fuchsian groups

1.4 Modular functions and modular forms

1.5 Congruence subgroups

1.6 Cusps of Г 0.(N) /H*

1.7 The normalizer of Г 0(N)

1.8 The genus of Г 0(N) / H*

1.9 The genus of Г H*, where Г= (Г0(N), We, Wf,)

1.10 The subgroup n\h + e, f

2. Dedekind eta function

2.1 The Dedekind eta function n(z)

2.2 The Poisson Sum Formula and applications

2.3 Theta transformation formula

2.4 Transformation formula for n(t)

2.5 Quadratic reciprocity law, quadratic characters, and Petersson constants

3 "Moonshine" of finite groups 31 3.1 Generalized partitions

3.2 Harmonies

3.3 Symmetric and alternating products of representations

4. Multiplicative product of n functions

Appendix. Genus zero discrete groups 65 Bibliography.

Notes:

Includes bibliographical references (pages 67-76).

Local Notes:

Acquired for the Penn Libraries with assistance from the Rosengarten Family Fund.

ISBN:

3037190906

9783037190906

OCLC:

671693342

Publisher Number:

99996450823