**University Lecture Series**

Volume: 15;
1999;
188 pp;
Softcover

MSC: Primary 20; 16;
Secondary 05

Print ISBN: 978-0-8218-1926-5

Product Code: ULECT/15

List Price: $30.00

AMS Member Price: $24.00

MAA member Price: $27.00

**Electronic ISBN: 978-1-4704-2164-9
Product Code: ULECT/15.E**

List Price: $30.00

AMS Member Price: $24.00

MAA member Price: $27.00

# Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group

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*Andrew Mathas*

This volume presents a fully self-contained introduction to
the modular representation theory of the Iwahori-Hecke 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 Iwahori-Hecke
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.

#### Readership

Graduate students and research mathematicians interested in group theory and generalizations; some physicists.

#### Reviews & Endorsements

Mathas' book contains many exercises which introduce the reader to a number of further interesting topics (e.g., the Robinson-Schnested 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

#### Table of Contents

# Table of Contents

## Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Introduction ix10 free
- Chapter 1. The Iwahori-Hecke algebra of the symmetric group 116 free
- Chapter 2. Cellular algebras 1530
- Chapter 3. The modular representation theory of H 2742
- Chapter 4. The q-Schur algebra 5570
- Chapter 5. The Jantzen sum formula and the blocks of H 6984
- Chapter 6. Branching rules, canonical bases and decomposition matrices 95110
- 1. The LLT algorithm 95110
- 2. Decomposition maps and adjustment matrices 115130
- 3. The Kleshchev-Brundan modular branching rules 118133
- 4. Rules for computing decomposition matrices 122137
- 5. The q-Schur algebras and GL[sub(n)](q) 129144
- 6. The Ariki-Koike algebras and cyclotomic q-Schur algebras 131146

- Appendix A. Finite dimensional algebras over a field 137152
- Appendix B. Decomposition matrices 149164
- Appendix C. Elementary divisors of integral Specht modules 165180
- Index of notation 177192 free
- References 181196
- Index 187202
- Back Cover Back Cover1204