Characters of Groups and Lattices over Orders: From Ordinary to Integral Representation Theory

Alexander Zimmermann

Published by De Gruyter 2022

Characters of Groups and Lattices over Orders

From Ordinary to Integral Representation Theory

Alexander Zimmermann

In the series De Gruyter Textbook

https://doi.org/10.1515/9783110702446

Overview
Contents

About this book

This is the fi rst textbook leading coherently from classical character theory to the theory of lattices over orders and integral representations of fi nite groups. Character theory is developed in a highly pedagogical way including many examples and exercises covering at once the fi rst defi nitions up to Clifford theory, Brauer’s induction theorem and the splitting fi eld theorem, as well as self-dual simple modules allowing bilinear forms. This latter part is done step by step using the approach given by Sin and Willems. Dirichlet characters and Dirichlet’s result on primes in arithmetic progressions are given as an application. Examples of integral representations of fi nite groups are already detailed at a quite early stage where appropriate, so that the more systematic treatment of lattices over orders is natural. After that, the necessary number theory and homological algebra is developed as far as needed. Maximal as well as hereditary orders are introduced and the Auslander- Buchsbaum theorem is proved. The Jordan-Zassenhaus theorem on the fi niteness of lattices in a given vector space is fully proved. Then the development and properties of class groups of orders is a fi rst focus. As a further highlight Swan’s example of a stably free but not free ideal over the integral group ring of the generalised quaternion group of order 32 is developed in great detail.

A student friendly introduction to ordinary representation theory
Many examples and exercises, including solutions for some of them, make the book well suited for self-study
Leads coherently from ordinary character theory to the integral representation theory of lattices over orders
Several topics appear for the fi rst time in a textbook, such as Sin-Willems’ approach to self-dual simple modules and Swan‘s example of a stably free non free ideal.

Introduction into some of the most exciting phenomena in number theory at a still elementary level.

Numerous exercises in all chapters.

Author / Editor information

Alexander Zimmermann, Université de Picardie, Amiens, France.

Topics

Frontmatter Publicly Available Download PDF	I
Preface Publicly Available Download PDF	VII
Contents Publicly Available Download PDF	XI
1 Ring theoretical foundations Requires Authentication Unlicensed Licensed Download PDF	1
2 Characters Requires Authentication Unlicensed Licensed Download PDF	45
3 Tensor products, Mackey formulas and Clifford theory Requires Authentication Unlicensed Licensed Download PDF	93
4 Bilinear forms on modules Requires Authentication Unlicensed Licensed Download PDF	139
5 Brauer induction, Brauer’s splitting field theorem Requires Authentication Unlicensed Licensed Download PDF	159
6 Some homological algebra methods in ring theory Requires Authentication Unlicensed Licensed Download PDF	183
7 Some algebraic number theory Requires Authentication Unlicensed Licensed Download PDF	223
8 Some notions of integral representations Requires Authentication Unlicensed Licensed Download PDF	265
9 Solution to selected exercises Requires Authentication Unlicensed Licensed Download PDF	339
Bibliography Requires Authentication Unlicensed Licensed Download PDF	349
Index Requires Authentication Unlicensed Licensed Download PDF	353

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Students and lecturers in mathematics.
Pages
- Front matter: 14
- Main content: 358
Illustrations
- Tables: 10
Keywords: Characters; Group representations; Quadratic modules; Dirichlet L-series; Lattices over orders

eBook

Published: January 19, 2022
ISBN: 9783110702446

Paperback

Published: January 31, 2022
ISBN: 9783110702439