Schur Algebras and Representation Theory

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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 algebras and related areas. He discusses the usual representation-theoretic topics such as constructions of irreducible modules, the blocks containing them, their modular characters and the problem of computing decomposition numbers; moreover deeper properties such as the quasi-hereditariness of the Schur algebra are discussed. The opportunity is taken to give an account of quantum versions of Schur algebras and their relations with certain q-deformations of the coordinate rings of the general linear group. The approach is combinatorial where possible, making the presentation accessible to graduate students. A few topics, however, require results from the representation theory of algebraic groups; so, to keep the book reasonably self-contained, an appendix on that is included. This is the first comprehensive text in this important and active area of research; it will be of interest to all research workers in representation theory.
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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: 9780521100465
  • Publisher: Cambridge University Press
  • Publication date: 1/18/2009
  • Series: Cambridge Tracts in Mathematics Series, #112
  • Pages: 256
  • Product dimensions: 6.00 (w) x 8.90 (h) x 0.80 (d)

Table of Contents

1 Polynomial functions and combinatorics 1
1.1 Introductory remarks 1
1.2 Schur's thesis 2
1.3 The polynomial algebra 9
1.4 Combinatorics 13
1.5 Character theory and weight spaces 17
1.6 Irreducible objects in P[subscript K]n, r 22
2 The Schur algebra 25
2.1 Definition 26
2.2 First properties 28
2.3 The Schur algebra S[subscript K]n,r 36
2.4 Bideterminants and codeterminants 38
2.5 The Straightening Formula 45
2.6 The Desarmenien matrix and independence 53
3 Representation theory of the Schur algebra 59
3.1 Modules for [Alpha subscript r] and S[subscript r] 60
3.2 Schur modules as induced modules 63
3.3 Heredity chains 67
3.4 Schur modules and Weyl modules 73
3.5 Modular representation theory for Schur algebras 84
4 Schur functors and the symmetric group 89
4.1 The Schur functor 90
4.2 Applying the Schur functor 94
4.3 Hom functors for quasi-hereditary algebras 107
4.4 Decomposition numbers for G and [Gamma] 111
4.5 [Delta]-[actual symbol not reproducible]-good filtrations 116
4.6 Young modules 122
5 Block theory 130
5.1 Summary of block theory 131
5.2 Return of the Hom functors 137
5.3 Primitive blocks 138
5.4 General blocks 143
5.5 The finiteness theorem 146
5.6 Examples 151
6 The q-Schur algebra 159
6.1 Quantum matrix space 161
6.2 The q-Schur algebra, first visit 164
6.3 Weights and polynomial modules 166
6.4 Characters and irreducible [Alpha subscript q]n-modules 168
6.5 R-forms for q-Schur algebras 170
6.6 The q-Schur algebra, second visit 173
7 Representation theory of S[subscript q]n,r 185
7.1 q-Weyl modules 186
7.2 The q-determinant in [Alpha subscript q]n,r 192
7.3 A quantum GL[subscript n] 195
7.4 The category P[subscript q]n,r 199
7.5 P[subscript q]n,r is a highest weight category 200
7.6 Representations of GL[subscript n]q and the q-Young modules 202
7.7 Conclusion 210
Appendix: a review of algebraic groups 212
A.1 Linear algebraic groups: definitions 212
A.2 Examples of linear algebraic groups 213
A.3 The weight lattice 214
A.4 Root systems 214
A.5 Weyl groups 215
A.6 The affine Weyl group 216
A.7 Simple modules for reductive groups 217
A.8 General linear group schemes 218
References 220
Index of Notation 227
Index of Terms 229
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