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Group Theory

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Book Id: WPLBN0000659316
Format Type: PDF eBook
File Size: 1.70 MB
Reproduction Date: 2005
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Title: Group Theory  
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Language: English
Subject: Science., Mathematics, Logic
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Citation

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Group Theory. (n.d.). Group Theory. Retrieved from http://hawaiilibrary.net/


Description
Mathematics document containing theorems and formulas.

Table of Contents
Contents 1 Introduction 1 2 Apreview 5 2.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 First example: SU(n) . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Second example: E6 family . . . . . . . . . . . . . . . . . . . . . . 12 3 Invariants and reducibility 15 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1.2 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1.3 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.4 Defining space, tensors, representations . . . . . . . . . . . 18 3.2 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 Algebra of invariants . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Invariance groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Projection operators . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5 Further invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 Birdtracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.7 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . 29 3.8 Zero- and one-dimensional subspaces . . . . . . . . . . . . . . . . . 31 3.9 Infinitesimal transformations . . . . . . . . . . . . . . . . . . . . . 32 3.10 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.11 Other forms of Lie algebra commutators . . . . . . . . . . . . . . . 38 3.12 Irrelevancy of clebsches . . . . . . . . . . . . . . . . . . . . . . . . 38 4 Recouplings 41 4.1 Couplings and recouplings . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Wigner 3n ? j coefficients . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . . 45 5 Permutations 49 5.1 Permutations in birdtracks . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Symmetrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.4 Levi-Civita tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.6 Characteristic equations . . . . . . . . . . . . . . . . . . . . . . . . 57 5.7 Fully (anti)symmetric tensors . . . . . . . . . . . . . . . . . . . . . 57 5.8 Young tableaux, Dynkin labels . . . . . . . . . . . . . . . . . . . . 58 6 Casimir operators 59 6.1 Casimirs and Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 Independent casimirs . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.3 Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.4 Dynkin indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.5 Quadratic, cubic casimirs . . . . . . . . . . . . . . . . . . . . . . . 60 6.6 Quartic casimirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.7 Sundry relations between quartic casimirs . . . . . . . . . . . . . . 60 6.8 Identically vanishing tensors . . . . . . . . . . . . . . . . . . . . . . 60 6.9 Dynkin labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7 Group integrals 63 7.1 Group integrals for arbitrary representations . . . . . . . . . . . . 64 7.2 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.3 Examples of group integrals . . . . . . . . . . . . . . . . . . . . . . 64 8 Unitary groups 65 8.1 Two-index tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.2 Three-index tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 66 8.3 Young tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8.3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 8.3.2 SU(n) Young tableaux . . . . . . . . . . . . . . . . . . . . . 69 8.3.3 Reduction of direct products . . . . . . . . . . . . . . . . . 70 8.4 Young projection operators . . . . . . . . . . . . . . . . . . . . . . 71 8.4.1 A dimension formula . . . . . . . . . . . . . . . . . . . . . . 72 8.4.2 Dimension as the number of strand colorings . . . . . . . . 73 8.5 Reduction of tensor products . . . . . . . . . . . . . . . . . . . . . 74 8.5.1 Three- and four-index tensors . . . . . . . . . . . . . . . . . 74 8.5.2 Basis vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.6 3-j symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.6.1 Evaluation by direct expansion . . . . . . . . . . . . . . . . 77 8.6.2 Application of the negative dimension theorem . . . . . . . 77 8.6.3 A sum rule for 3-j?s . . . . . . . . . . . . . . . . . . . . . . 78 8.7 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.8 Mixed two-index tensors . . . . . . . . . . . . . . . . . . . . . . . . 79 8.9 Mixed defining ? adjoint tensors . . . . . . . . . . . . . . . . . . . 81 8.10 Two-index adjoint tensors . . . . . . . . . . . . . . . . . . . . . . . 83

 

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