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Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group
Andrew Mathas, University of Sydney, NSW, Australia
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University Lecture Series
1999; 188 pp; softcover
Volume: 15
ISBN-10: 0-8218-1926-7
ISBN-13: 978-0-8218-1926-5
List Price: US$28 Member Price: US$22.40
Order Code: ULECT/15

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.

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

Reviews

"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

• The Iwahori-Hecke algebra of the symmetric group
• Cellular algebras
• The modular representation theory of $$\mathcal{H}$$
• The $$q$$-Schur algebra
• The Jantzen sum formula and the blocks of $$\mathcal H$$
• 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
• Index of notation
• References
• Index