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Why Group Theory? | |

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Finite Groups | |

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Groups and representations | |

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Example - Z[subscript 3] | |

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The regular representation | |

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Irreducible representations | |

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Transformation groups | |

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Application: parity in quantum mechanics | |

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Example: S[subscript 3] | |

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Example: addition of integers | |

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Useful theorems | |

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Subgroups | |

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Schur's lemma | |

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* Orthogonality relations | |

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Characters | |

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Eigenstates | |

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Tensor products | |

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Example of tensor products | |

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* Finding the normal modes | |

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* Symmetries of 2n+1-gons | |

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Permutation group on n objects | |

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Conjugacy classes | |

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Young tableaux | |

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Example -- our old friend S[subscript 3] | |

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Another example -- S[subscript 4] | |

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* Young tableaux and representations of S[subscript n] | |

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Lie Groups | |

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Generators | |

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Lie algebras | |

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The Jacobi identity | |

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The adjoint representation | |

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Simple algebras and groups | |

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States and operators | |

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Fun with exponentials | |

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SU(2) | |

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J[subscript 3] eigenstates | |

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Raising and lowering operators | |

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The standard notation | |

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Tensor products | |

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J[subscript 3] values add | |

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Tensor Operators | |

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Orbital angular momentum | |

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Using tensor operators | |

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The Wigner-Eckart theorem | |

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Example | |

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* Making tensor operators | |

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Products of operators | |

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Isospin | |

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Charge independence | |

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Creation operators | |

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Number operators | |

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Isospin generators | |

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Symmetry of tensor products | |

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The deuteron | |

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Superselection rules | |

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Other particles | |

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Approximate isospin symmetry | |

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Perturbation theory | |

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Roots and Weights | |

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Weights | |

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More on the adjoint representation | |

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Roots | |

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Raising and lowering | |

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Lots of SU(2)s | |

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Watch carefully - this is important! | |

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SU(3) | |

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The Gell-Mann matrices | |

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Weights and roots of SU(3) | |

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Simple Roots | |

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Positive weights | |

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Simple roots | |

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Constructing the algebra | |

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Dynkin diagrams | |

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Example: G[subscript 2] | |

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The roots of G[subscript 2] | |

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The Cartan matrix | |

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Finding all the roots | |

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The SU(2)s | |

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Constructing the G[subscript 2] algebra | |

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Another example: the algebra C[subscript 3] | |

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Fundamental weights | |

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The trace of a generator | |

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More SU(3) | |

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Fundamental representations of SU(3) | |

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Constructing the states | |

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The Weyl group | |

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Complex conjugation | |

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Examples of other representations | |

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Tensor Methods | |

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Lower and upper indices | |

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Tensor components and wave functions | |

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Irreducible representations and symmetry | |

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Invariant tensors | |

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Clebsch-Gordan decomposition | |

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Triality | |

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Matrix elements and operators | |

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Normalization | |

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Tensor operators | |

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The dimension of (n,m) | |

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* The weights of (n,m) | |

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Generalization of Wigner-Eckart | |

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* Tensors for SU(2) | |

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* Clebsch-Gordan coefficients from tensors | |

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* Spin s[subscript 1] + s[subscript 2] - 1 | |

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* Spin s[subscript 1] + s[subscript 2] - k | |

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Hypercharge and Strangeness | |

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The eight-fold way | |

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The Gell-Mann Okubo formula | |

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Hadron resonances | |

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Quarks | |

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Young Tableaux | |

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Raising the indices | |

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Clebsch-Gordan decomposition | |

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SU(3) [right arrow] SU(2) [times] U(1) | |

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SU(N) | |

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Generalized Gell-Mann matrices | |

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SU(N) tensors | |

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Dimensions | |

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Complex representations | |

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SU(N) [multiply sign in circle] SU(M) [set membership] SU(N +M) | |

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3-D Harmonic Oscillator | |

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Raising and lowering operators | |

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Angular momentum | |

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A more complicated example | |

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SU(6) and the Quark Model | |

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Including the spin | |

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SU(N) [multiply sign in circle] SU(M) [set membership] SU(NM) | |

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The baryon states | |

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Magnetic moments | |

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Color | |

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Colored quarks | |

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Quantum Chromodynamics | |

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Heavy quarks | |

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Flavor SU(4) is useless! | |

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Constituent Quarks | |

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The nonrelativistic limit | |

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Unified Theories and SU(5) | |

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Grand unification | |

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Parity violation, helicity and handedness | |

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Spontaneously broken symmetry | |

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Physics of spontaneous symmetry breaking | |

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Is the Higgs real? | |

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Unification and SU(5) | |

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Breaking SU(5) | |

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Proton decay | |

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The Classical Groups | |

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The SO(2n) algebras | |

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The SO(2n + 1) algebras | |

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The Sp(2n) algebras | |

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Quaternions | |

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The Classification Theorem | |

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II-systems | |

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Regular subalgebras | |

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Other Subalgebras | |

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SO(2n + 1) and Spinors | |

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Fundamental weight of SO(2n + 1) | |

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Real and pseudo-real | |

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Real representations | |

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Pseudo-real representations | |

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R is an invariant tensor | |

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The explicit form for R | |

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SO(2n + 2) Spinors | |

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Fundamental weights of SO(2n + 2) | |

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SU(n) [subset or is implied by] SO(2n) | |

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Clifford algebras | |

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[Gamma][subscript m] and R as invariant tensors | |

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Products of [Gamma][subscript s] | |

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Self-duality | |

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Example: SO(10) | |

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The SU(n) subalgebra | |

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SO(10) | |

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SO(10) and SU(4) [times] SU(2) [times] SU(2) | |

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* Spontaneous breaking of SO(10) | |

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* Breaking SO(10) [right arrow] SU(5) | |

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* Breaking SO(10) [right arrow] SU(3) [times] SU(2) [times] U(1) | |

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* Breaking SO(10) [right arrow] SU(3) [times] U(1) | |

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* Lepton number as a fourth color | |

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Automorphisms | |

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Outer automorphisms | |

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Fun with SO(8) | |

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Sp(2n) | |

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Weights of SU(n) | |

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Tensors for Sp(2n) | |

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Odds and Ends | |

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Exceptional algebras and octonians | |

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E[subscript 6] unification | |

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Breaking E[subscript 6] | |

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SU(3) [times] SU(3) [times] SU(3) unification | |

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Anomalies | |

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Epilogue | |

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Index | |