Characters of Groups and Lattices over Orders : : From Ordinary to Integral Representation Theory / / Alexander Zimmermann.
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 nition...
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Place / Publishing House: | Berlin ;, Boston : : De Gruyter, , [2022] ©2022 |
Year of Publication: | 2022 |
Language: | English |
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Zimmermann, Alexander, author. aut http://id.loc.gov/vocabulary/relators/aut Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / Alexander Zimmermann. Berlin ; Boston : De Gruyter, [2022] ©2022 1 online resource (XIV, 358 p.) text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda De Gruyter Textbook Frontmatter -- Preface -- Contents -- 1 Ring theoretical foundations -- 2 Characters -- 3 Tensor products, Mackey formulas and Clifford theory -- 4 Bilinear forms on modules -- 5 Brauer induction, Brauer’s splitting field theorem -- 6 Some homological algebra methods in ring theory -- 7 Some algebraic number theory -- 8 Some notions of integral representations -- 9 Solution to selected exercises -- Bibliography -- Index restricted access http://purl.org/coar/access_right/c_16ec online access with authorization star 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. Issued also in print. Mode of access: Internet via World Wide Web. In English. Description based on online resource; title from PDF title page (publisher's Web site, viewed 02. Mai 2023) Dirichlet-L-Reihen. Gruppendarstellungen. Quadratische Formen. MATHEMATICS / Group Theory. bisacsh Characters. Dirichlet L-series. Group representations. Lattices over orders. Quadratic modules. Title is part of eBook package: De Gruyter DG Plus DeG Package 2022 Part 1 9783110766820 Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2022 English 9783110993899 Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2022 9783110994810 ZDB-23-DGG Title is part of eBook package: De Gruyter EBOOK PACKAGE Mathematics 2022 English 9783110993868 Title is part of eBook package: De Gruyter EBOOK PACKAGE Mathematics 2022 9783110770445 ZDB-23-DMA EPUB 9783110702552 print 9783110702439 https://doi.org/10.1515/9783110702446 https://www.degruyter.com/isbn/9783110702446 Cover https://www.degruyter.com/document/cover/isbn/9783110702446/original |
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author |
Zimmermann, Alexander, Zimmermann, Alexander, |
spellingShingle |
Zimmermann, Alexander, Zimmermann, Alexander, Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / De Gruyter Textbook Frontmatter -- Preface -- Contents -- 1 Ring theoretical foundations -- 2 Characters -- 3 Tensor products, Mackey formulas and Clifford theory -- 4 Bilinear forms on modules -- 5 Brauer induction, Brauer’s splitting field theorem -- 6 Some homological algebra methods in ring theory -- 7 Some algebraic number theory -- 8 Some notions of integral representations -- 9 Solution to selected exercises -- Bibliography -- Index |
author_facet |
Zimmermann, Alexander, Zimmermann, Alexander, |
author_variant |
a z az a z az |
author_role |
VerfasserIn VerfasserIn |
author_sort |
Zimmermann, Alexander, |
title |
Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / |
title_sub |
From Ordinary to Integral Representation Theory / |
title_full |
Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / Alexander Zimmermann. |
title_fullStr |
Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / Alexander Zimmermann. |
title_full_unstemmed |
Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / Alexander Zimmermann. |
title_auth |
Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / |
title_alt |
Frontmatter -- Preface -- Contents -- 1 Ring theoretical foundations -- 2 Characters -- 3 Tensor products, Mackey formulas and Clifford theory -- 4 Bilinear forms on modules -- 5 Brauer induction, Brauer’s splitting field theorem -- 6 Some homological algebra methods in ring theory -- 7 Some algebraic number theory -- 8 Some notions of integral representations -- 9 Solution to selected exercises -- Bibliography -- Index |
title_new |
Characters of Groups and Lattices over Orders : |
title_sort |
characters of groups and lattices over orders : from ordinary to integral representation theory / |
series |
De Gruyter Textbook |
series2 |
De Gruyter Textbook |
publisher |
De Gruyter, |
publishDate |
2022 |
physical |
1 online resource (XIV, 358 p.) Issued also in print. |
contents |
Frontmatter -- Preface -- Contents -- 1 Ring theoretical foundations -- 2 Characters -- 3 Tensor products, Mackey formulas and Clifford theory -- 4 Bilinear forms on modules -- 5 Brauer induction, Brauer’s splitting field theorem -- 6 Some homological algebra methods in ring theory -- 7 Some algebraic number theory -- 8 Some notions of integral representations -- 9 Solution to selected exercises -- Bibliography -- Index |
isbn |
9783110702446 9783110766820 9783110993899 9783110994810 9783110993868 9783110770445 9783110702552 9783110702439 |
url |
https://doi.org/10.1515/9783110702446 https://www.degruyter.com/isbn/9783110702446 https://www.degruyter.com/document/cover/isbn/9783110702446/original |
illustrated |
Not Illustrated |
doi_str_mv |
10.1515/9783110702446 |
oclc_num |
1290486397 |
work_keys_str_mv |
AT zimmermannalexander charactersofgroupsandlatticesoverordersfromordinarytointegralrepresentationtheory |
status_str |
n |
ids_txt_mv |
(DE-B1597)549627 (OCoLC)1290486397 |
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cr |
hierarchy_parent_title |
Title is part of eBook package: De Gruyter DG Plus DeG Package 2022 Part 1 Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2022 English Title is part of eBook package: De Gruyter EBOOK PACKAGE COMPLETE 2022 Title is part of eBook package: De Gruyter EBOOK PACKAGE Mathematics 2022 English Title is part of eBook package: De Gruyter EBOOK PACKAGE Mathematics 2022 |
is_hierarchy_title |
Characters of Groups and Lattices over Orders : From Ordinary to Integral Representation Theory / |
container_title |
Title is part of eBook package: De Gruyter DG Plus DeG Package 2022 Part 1 |
_version_ |
1770177778482675712 |
fullrecord |
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