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| Preface: the pursuit of symmetries | |
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| Finite groups: an introduction | |
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| Group axioms | |
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| Finite groups of low order | |
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| Permutations | |
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| Basic concepts | |
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| Conjugation | |
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| Simple groups | |
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| Sylow's criteria | |
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| Semi-direct product | |
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| Young Tableaux | |
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| Finite groups: representations | |
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| Introduction | |
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| Schur's lemmas | |
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| The A<sub>4</sub> character table | |
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| Kronecker products | |
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| Real and complex representations | |
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| Embeddings | |
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| Z<sub>n</sub> character table | |
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| D<sub>n</sub> character table | |
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| Q<sub>2n</sub> character table | |
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| Some semi-direct products | |
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| Induced representations | |
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| Invariants | |
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| Coverings | |
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| Hilbert spaces | |
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| Finite Hilbert spaces | |
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| Fermi oscillators | |
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| Infinite Hilbert spaces | |
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| SU (2) | |
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| Introduction | |
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| Some representations | |
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| From Lie algebras to Lie groups | |
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| SU (2) → SU (1, 1) | |
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| Selected SU (2) applications | |
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| The isotropic harmonic oscillator | |
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| The Bohr atom | |
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| Isotopic spin | |
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| SU (3) | |
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| SU (3) algebra | |
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| �-Basis | |
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| �-Basis | |
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| �′-Basis | |
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| The triplet representation | |
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| The Chevalley basis | |
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| SU (3) in physics | |
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| The isotropic harmonic oscillator redux | |
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| The Elliott model | |
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| The Sakata model | |
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| The Eightfold Way | |
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| Classification of compact simple Lie algebras | |
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| Classification | |
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| Simple roots | |
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| Rank-two algebras | |
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| Dynkin diagrams | |
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| Orthonormal bases | |
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| Lie algebras: representation theory | |
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| Representation basics | |
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| A<sub>3</sub> fundamentals | |
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| The Weyl group | |
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| Orthogonal Lie algebras | |
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| Spinor representations | |
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| SO (2n) spinors | |
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| SO (2n + 1) spinors | |
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| Clifford algebra construction | |
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| Casimir invariants and Dynkin indices | |
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| Embeddings | |
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| Oscillator representations | |
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| Verma modules | |
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| Weyl dimension formula | |
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| Verma basis | |
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| Finite groups: the road to simplicity | |
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| Matrices over Galois fields | |
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| PSL<sub>2</sub> (7) | |
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| A doubly transitive group | |
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| Chevalley groups | |
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| A fleeting glimpse at the sporadic groups | |
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| Beyond Lie algebras | |
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| Serre presentation | |
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| Affine Kac-Moody algebras | |
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| Super algebras | |
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| The groups of the Standard Model | |
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| Space-time symmetries | |
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| The Lorentz and Poincar� groups | |
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| The conformal group | |
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| Beyond space-time symmetries | |
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| Color and the quark model | |
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| Invariant Lagrangians | |
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| Non-Abelian gauge theories | |
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| The Standard Model | |
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| Grand Unification | |
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| Possible family symmetries | |
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| Finite SU (2) and SO (3) subgroups | |
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| Finite SU (3) subgroups | |
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| Exceptional structures | |
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| Hurwitz algebras | |
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| Matrices over Hurwitz algebras | |
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| The Magic Square | |
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| Properties of some finite groups | |
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| Properties of selected Lie algebras | |
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| References | |
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| Index | |