Polynomial Representations of the General Linear Group

Contents......Page 3
Introduction......Page 6
1.1 The Definition of a Coalgebra......Page 8
1.1.1 Examples of Coalgebras......Page 9
1.2 The Dual Algebra to a Coalgebra......Page 10
1.3 Homomorphisms of Coalgebras......Page 11
1.5 Comodules......Page 13
1.6 Bialgebras......Page 15
1.7 Definitions in Module Theory......Page 16
1.7.2 Absolute Irreducibility......Page 17
2.1 Basic Representation Theory......Page 18
2.2.1 F is a K-coalgebra......Page 19
2.3 Coefficient Functions......Page 22
2.4 The category modA(K)......Page 23
3 Polynomial Representations  and the Schur Algebra......Page 24
3.1 The Definition of MK(n) and MK(n,r)......Page 26
3.2 Examples of Polynomial Representations......Page 27
3.3 The Schur Algebra......Page 28
3.4 The map e : KSK(n,r)......Page 30
3.5 The Module Er......Page 32
4.1 Weights......Page 35
4.2 Weight Spaces......Page 36
4.2.1 Examples of Weight Spaces......Page 38
4.3 First Results on Weight Spaces......Page 39
4.4 Characters......Page 40
4.5 Irreducible modules in MK(n,r)......Page 45
5.1 Conjugacy Classes of GL2(Fq)......Page 48
5.2 Irreducible Characters of V,U and V......Page 50
5.2.1 The Characters of U and V......Page 52
5.3 The Characters of W,......Page 53
5.4 The Characters of `3́9`42`\"̇613A``45`47`\"603AInd......Page 56
Conclusion......Page 61
Acknowledgements......Page 62