Lectures in mathematics : Kyoto University

vol. 12

Theory of group characters / by Richard Brauer ; notes prepared by T. Tsuzuku in cooperation with T. Nakayama ... [et al.]
c1979

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I. Preliminaries
§ 1. Algebras and their representations 1-8
§ 2. Representations of finite groups - classical theory 8-19
§ 3. Cyclotomic splitting fields 19-30
II. Arithmetical Structure
§ 1. The numbers ω_i(K_α) 31
§ 2. Modular representations and their characters 32-33
§ 3. Transition from an ordinary representation to a modular one 33-34
§ 4. Decomposition numbers and Cartan invariants 34-37
§ 5. The number of irreducible modular characters 37-39
§ 6. The chacacters φ_ρ 39-41
§ 7. The p-rank of the matrix D of decomposition numbers 41-43
§ 8. Blocks 43-46
§ 9. Idempotents belonging to blocks 46-50
§ 10. Defect of a block 50-52
§ 11. The determinant of the matrix C of Cartan invariants 52-56
§ 12. The elementary divisors of C and C_τ 56-63
§ 13. The number of irreducible characters in a block 63-66
III. Defect groups. Main theorem A
§ 1. Blocks of G and those of subgroups 67-71
§ 2. Defect groups of a block 71-78
§ 3. Main Theorem A 78-81
IV. Generalized Decomposition Numbers, Main Theorem B
§ 1. Generalized decomposition numbers 82-84
§ 2. Sections 84-88
§ 3. The matrix of generalized decomposition numbers 88-92
§ 4. Main Theorem B 93
§ 5. Some consequences 93-104
References 104-107

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