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Linear and Projective Representations of Symmetric Groups

$70.99 (C)

Part of Cambridge Tracts in Mathematics

  • Date Published: March 2009
  • availability: Available
  • format: Paperback
  • isbn: 9780521104180

$ 70.99 (C)

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About the Authors
  • The representation theory of symmetric groups is one of the most beautiful, popular, and important parts of algebra with many deep relations to other areas of mathematics, such as combinatorics, Lie theory, and algebraic geometry. Kleshchev describes a new approach to the subject, based on the recent work of Lascoux, Leclerc, Thibon, Ariki, Grojnowski, Brundan, and the author. Much of this work has only appeared in the research literature before. However, to make it accessible to graduate students, the theory is developed from scratch, the only prerequisite being a standard course in abstract algebra. Branching rules are built in from the outset resulting in an explanation and generalization of the link between modular branching rules and crystal graphs for affine Kac-Moody algebras. The methods are purely algebraic, exploiting affine and cyclotomic Hecke algebras. For the first time in book form, the projective (or spin) representation theory is treated along the same lines as linear representation theory. The author is mainly concerned with modular representation theory, although everything works in arbitrary characteristic, and in case of characteristic 0 the approach is somewhat similar to the theory of Okounkov and Vershik, described here in chapter 2. For the sake of transparency, Kleshschev concentrates on symmetric and spin-symmetric groups, though the methods he develops are quite general and apply to a number of related objects. In sum, this unique book will be welcomed by graduate students and researchers as a modern account of the subject.

    Reviews & endorsements

    "The book is written with great care and in a dense style... The author has mastered a very difficult task in writing this book and has enriched the literature on the symmetric groups with a unique and very valuable monograph, making the formidable recent developments more widely accessible by starting the presentation from scratch."
    Christine Bessenrodt, Mathematical Reviews

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    Product details

    • Date Published: March 2009
    • format: Paperback
    • isbn: 9780521104180
    • length: 292 pages
    • dimensions: 229 x 152 x 17 mm
    • weight: 0.43kg
    • availability: Available
  • Table of Contents

    Part I. Linear Representations:
    1. Notion and generalities
    2. Symmetric groups I
    3. Degenerate affine Hecke algebra
    4. First results on Hn modules
    5. Crystal operators
    6. Character calculations
    7. Integral representations and cyclotomic Hecke algebras
    8. Functors e and f
    9. Construction of Uz and irreducible modules
    10. Identification of the crystal
    11. Symmetric groups II
    Part II. Projective Representations:
    12. Generalities on superalgebra
    13. Sergeev superalgebras
    14. Affine Sergeev superalgebras
    15. Integral representations and cyclotomic Sergeev algebras
    16. First results on Xn modules
    17. Crystal operators fro Xn
    18. Character calculations for Xn
    19. Operators e and f
    20. Construction of Uz and irreducible modules
    21. Identification of the crystal
    22. Double covers

  • Author

    Alexander Kleshchev, University of Oregon
    Alexander Kleshchev is a Professor of Mathematics at the University of Oregon.

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