Schur algebras and representation theory [E-Book] / Stuart Martin.
Martin, Stuart, (author)
Cambridge : Cambridge University Press, 1993
1 online resource (xv, 232 pages)
englisch
9780521100465
9780511470899
9780521415910
Cambridge tracts in mathematics ; 112
Full Text
Table of 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.