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
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