Schur algebras and representation theory / Stuart Martin.

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The Schur algebra is an algebraic system providing a link between the representation theory of the symmetric and general linear groups (both finite and infinite). In the text Dr Martin gives a full, self-contained account of this algebra and these links, covering both the basic theory of Schur algeb...

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Bibliographic Details
Uniform Title:Cambridge tracts in mathematics ; 112.
Main Author: Martin, Stuart, 1964- (Author)
Language:English
Published: Cambridge : Cambridge University Press, 1993.
Series:Cambridge tracts in mathematics ; 112.
Subjects:
Representations of groups.
Representations of algebras.
Online Access:
Connect to online resource - MSU authorized users (Digital Object Identifier Permalink)
Physical Description:1 online resource (xv, 232 pages) : digital, PDF file(s).
Variant Title:
Schur Algebras & Representation Theory.
Format: Electronic eBook
Contents:
  • 1. Polynomial functions and combinatorics. 1.1. Introductory remarks. 1.2. Schur's thesis. 1.3. The polynomial algebra. 1.4. Combinatorics. 1.5. Character theory and weight spaces. 1.6. Irreducible objects in P[subscript K](n, r)
  • 2. The Schur algebra. 2.1. Definition. 2.2. First properties. 2.3. The Schur algebra S[subscript K](n, r). 2.4. Bideterminants and codeterminants. 2.5. The Straightening Formula. 2.6. The Desarmenien matrix and independence
  • 3. Representation theory of the Schur algebra. 3.1. Modules for [Alpha subscript r] and S[subscript r]. 3.2. Schur modules as induced modules. 3.3. Heredity chains. 3.4. Schur modules and Weyl modules. 3.5. Modular representation theory for Schur algebras
  • 4. Schur functors and the symmetric group. 4.1. The Schur functor. 4.2. Applying the Schur functor. 4.3. Hom functors for quasi-hereditary algebras. 4.4. Decomposition numbers for G and [Gamma]. 4.5. [Delta]-[actual symbol not reproducible]-good filtrations. 4.6. Young modules
  • 5. Block theory.
  • 5.1. Summary of block theory. 5.2. Return of the Hom functors. 5.3. Primitive blocks. 5.4. General blocks. 5.5. The finiteness theorem. 5.6. Examples
  • 6. The q-Schur algebra. 6.1. Quantum matrix space. 6.2. The q-Schur algebra, first visit. 6.3. Weights and polynomial modules. 6.4. Characters and irreducible [Alpha subscript q](n)-modules. 6.5. R-forms for q-Schur algebras. 6.6. The q-Schur algebra, second visit
  • 7. Representation theory of S[subscript q](n, r). 7.1. q-Weyl modules. 7.2. The q-determinant in [Alpha subscript q](n, r). 7.3. A quantum GL[subscript n]. 7.4. The category P[subscript q](n, r). 7.5. P[subscript q](n, r) is a highest weight category. 7.6. Representations of GL[subscript n](q) and the q-Young modules. 7.7. Conclusion
  • Appendix: a review of algebraic groups
  • A.1 Linear algebraic groups: definitions
  • A.2 Examples of linear algebraic groups
  • A.3 The weight lattice
  • A.4 Root systems
  • A.5 Weyl groups
  • A.6 The affine Weyl group.
  • A.7 Simple modules for reductive groups
  • A.8 General linear group schemes.