Schur Algebras and Representation Theory

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Schur algebras are an algebraic system that provide a link between the representation theory of the symmetric and general linear groups. Dr. Martin gives a self-contained account of this algebra and those links, covering the basic ideas and their quantum analogues. He discusses not only the usual representation-theoretic topics (such as constructions of irreducible modules, the structure of blocks containing them, decomposition numbers and so on) but also the intrinsic properties of Schur algebras, leading to a discussion of their cohomology theory. He also investigates the relationship between Schur algebras and other algebraic structures. Throughout, the approach uses combinatorial language where possible, thereby making the presentation accessible to graduate students. Some topics require results from algebraic group theory, which are contained in an appendix.

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Editorial Reviews

From the Publisher
"this book can be a source of useful information for beginners in the field of representations of symmetric and general linear groups and finite-dimensional algebras and specialists as well. Most of the treatments are combinatorial, so it is accessible to graduate students...the text is comprehensible and the research area is important and active...the book is readable and will be a handy book for specialists whose interests lie in this area." Jie Du, Mathematical Reviews

"An excellent and thorough survey of one of the currently liveliest topics in algebra. Congratulations for work well done, Mr, Martin." The Bulletin of Mathematics

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Product Details

  • ISBN-13: 9780521415910
  • Publisher: Cambridge University Press
  • Publication date: 2/28/2007
  • Series: Cambridge Tracts in Mathematics Series, #112
  • Pages: 252
  • Product dimensions: 5.98 (w) x 8.98 (h) x 0.67 (d)

Table of Contents

Introduction; 1. Polynomial functions and combinatorics; 2. The Schur algebra; 3. Representation theory of the Schur algebra; 4. Schur functors and the symmetric group; 5. Block theory; 6. The q-Schur algebra; 7. Representation theory of Sq (n, r); Appendix: a review of algebraic groups; References; Indexes.

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