vi CONTENTS

Chapter 10. The Hayashi realization 75

10.1. Partitions and the Hayashi realization 75

10.2. Generalization to the case of multipartitions 85

Chapter 11. Description of the crystal graph of V (Λ) 87

11.1. A theorem for proving the Misra-Miwa theorem 87

11.2. The Misra-Miwa theorem 94

Chapter 12. An overview of the applications to Hecke algebras 97

12.1. The Hecke algebra of type G(m, 1,n) 97

12.2. Consequences of Theorem 12.5 101

Chapter 13. The Hecke algebra of type G(m, 1,n) 105

13.1. The aﬃne Hecke algebra 105

13.2. Semi-normal representations 107

13.3. The decomposition map 111

13.4. Specht module theory 113

13.5. A theorem of Morita type 117

13.6. Refined induction and restriction functors 118

Chapter 14. The proof of Theorem 12.5 123

14.1. Representations of a cyclic quiver 123

14.2. The Hall algebra and the quantum algebra 127

14.3. Some results from the geometric theory 133

14.4. Proof of the generalized LLT conjecture 142

Chapter 15. Reference guide 147

Bibliography 149

Index 157