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Softcover ISBN:  9780821819265 
Product Code:  ULECT/15 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470421649 
Product Code:  ULECT/15.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9780821819265 
eBook ISBN:  9781470421649 
Product Code:  ULECT/15.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

Book DetailsUniversity Lecture SeriesVolume: 15; 1999; 188 ppMSC: Primary 20; 16; Secondary 05
This volume presents a fully selfcontained introduction to the modular representation theory of the IwahoriHecke algebras of the symmetric groups and of the \(q\)Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems.
The main results are proved by showing that the IwahoriHecke algebras and \(q\)Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in Chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the \(q\)Schur algebras.
This book is the first of its kind covering the topic. It offers a substantially simplified treatment of the original proofs. The book is a solid reference source for experts. It will also serve as a good introduction to students and beginning researchers since each chapter contains exercises and there is an appendix containing a quick development of the representation theory of algebras. A second appendix gives tables of decomposition numbers.
ReadershipGraduate students and research mathematicians interested in group theory and generalizations; some physicists.

Table of Contents

Chapters

Chapter 1. The IwahoriHecke algebra of the symmetric group

Chapter 2. Cellular algebras

Chapter 3. The modular representation theory of $\mathcal {H}$

Chapter 4. The $q$Schur algebra

Chapter 5. The Jantzen sum formula and the blocks of $\mathcal {H}$

Chapter 6. Branching rules, canonical bases and decomposition matrices

Appendix A. Finite dimensional algebras over a field

Appendix B. Decomposition matrices

Appendix C. Elementary divisors of integral Specht modules


Additional Material

Reviews

Mathas' book contains many exercises which introduce the reader to a number of further interesting topics (e.g., the RobinsonSchnested correspondence). Thus, students will find it useful. Historical notes at the end of each chapter provide some context for the discussion.
Bulletin of the AMS


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This volume presents a fully selfcontained introduction to the modular representation theory of the IwahoriHecke algebras of the symmetric groups and of the \(q\)Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems.
The main results are proved by showing that the IwahoriHecke algebras and \(q\)Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in Chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the \(q\)Schur algebras.
This book is the first of its kind covering the topic. It offers a substantially simplified treatment of the original proofs. The book is a solid reference source for experts. It will also serve as a good introduction to students and beginning researchers since each chapter contains exercises and there is an appendix containing a quick development of the representation theory of algebras. A second appendix gives tables of decomposition numbers.
Graduate students and research mathematicians interested in group theory and generalizations; some physicists.

Chapters

Chapter 1. The IwahoriHecke algebra of the symmetric group

Chapter 2. Cellular algebras

Chapter 3. The modular representation theory of $\mathcal {H}$

Chapter 4. The $q$Schur algebra

Chapter 5. The Jantzen sum formula and the blocks of $\mathcal {H}$

Chapter 6. Branching rules, canonical bases and decomposition matrices

Appendix A. Finite dimensional algebras over a field

Appendix B. Decomposition matrices

Appendix C. Elementary divisors of integral Specht modules

Mathas' book contains many exercises which introduce the reader to a number of further interesting topics (e.g., the RobinsonSchnested correspondence). Thus, students will find it useful. Historical notes at the end of each chapter provide some context for the discussion.
Bulletin of the AMS