Schur Algebras and Representation Theory Book

	Introduction
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