| Notation-Conventions-How to Use This Book | p. 1 |
| Introduction | p. 9 |
| The Rodrigues programme | p. 18 |
| Rotations by 2[pi] | p. 22 |
| Spinor representations | p. 24 |
| All You Need to Know About Symmetries, Matrices, and Groups | p. 29 |
| Symmetry operators in configuration space | p. 29 |
| Description of the point-symmetry operations | p. 30 |
| Specification of the symmetry operations | p. 32 |
| Composition of symmetry operations | p. 33 |
| Eigenvectors of configuration space operators | p. 35 |
| Symmetry operators in function space | p. 36 |
| Matrices and operators | p. 38 |
| Groups | p. 42 |
| All about matrix properties | p. 50 |
| Orthogonal matrices | p. 52 |
| Unitary matrices | p. 55 |
| Hermitian and skew-Hermitian matrices | p. 56 |
| Supermatrices and the direct product | p. 56 |
| Commutation of matrices | p. 58 |
| Matrix functions | p. 58 |
| Quantal symmetry. Observables and infinitesimal operators | p. 60 |
| Symmetries and observables | p. 62 |
| Infinitesimal operators and observables | p. 63 |
| A Primer on Rotations and Rotation Matrices | p. 65 |
| Euler angles | p. 66 |
| Rotation matrices in terms of the Euler angles | p. 69 |
| Angle and axis of an orthogonal matrix | p. 70 |
| The matrix of a rotation R([phi]n) | p. 73 |
| Euler angles in terms of the angle and axis of rotation | p. 75 |
| A rotation in terms of rotations about orthogonal axes | p. 76 |
| Comments on the parametrization of rotations | p. 79 |
| Rotations and Angular Momentum | p. 80 |
| Infinitesimal rotations | p. 80 |
| The infinitesimal generator: angular momentum | p. 82 |
| Rotation matrices | p. 84 |
| Commutation | p. 85 |
| Shift operators | p. 86 |
| The eigenfunctions of I[subscript z] | p. 87 |
| The irreducible bases for SO(3) | p. 90 |
| Spherical and solid harmonics | p. 93 |
| The Condon and Shortley convention | p. 94 |
| Applications. Matrices for j = 1 and j = 1/2 (Pauli matrices) | p. 96 |
| The Pauli matrices, j = 1/2 | p. 98 |
| Tensor Bases: Introduction to Spinors | p. 99 |
| Vectors and spherical vectors | p. 100 |
| Tensor bases and tensor products | p. 103 |
| Symmetrization of tensors | p. 104 |
| Half-integral bases: spinors | p. 106 |
| The Bilinear Transformation: Introduction to SU(2), SU'(2), and Rotations. More About Spinors | p. 109 |
| The bilinear transformation | p. 110 |
| The inverse | p. 111 |
| Special unitary matrices. The SU(2) group | p. 112 |
| Rotations and SU(2): a first contact | p. 113 |
| Binary rotations as the group generators | p. 116 |
| Do we have a representation of SO(3)? | p. 117 |
| SU(2) plus the inversion: SU'(2) | p. 118 |
| Inversion and parity | p. 120 |
| Spinors and their invariants | p. 121 |
| Rotations and SU(2). The Stereographic Projection | p. 124 |
| The stereographic projection | p. 124 |
| Geometry and coordinates of the projection | p. 126 |
| The homomorphism between SU(2) and SO(3) | p. 128 |
| The spinor components | p. 132 |
| Projective Representations | p. 135 |
| The group D[subscript 2] and its SU(2) matrices. Definition of projective representations | p. 135 |
| Bases of the projective representations | p. 138 |
| Bases and energy levels | p. 140 |
| The factor system | p. 142 |
| The representations | p. 144 |
| Characters | p. 144 |
| Direct products of representations | p. 145 |
| The covering group | p. 146 |
| Remarks | p. 149 |
| The Geometry of Rotations | p. 151 |
| The unit sphere and the rotation poles | p. 152 |
| Conjugate poles | p. 154 |
| Improper rotations | p. 154 |
| The Euler construction | p. 155 |
| Spherical trigonometry revisited | p. 157 |
| The Euler construction in formulae. The Euler-Rodrigues parameters | p. 159 |
| Remarks | p. 160 |
| The conical transformation | p. 162 |
| The Topology of Rotations | p. 164 |
| The parametric ball | p. 165 |
| Paths | p. 167 |
| Programme: continuity | p. 170 |
| Homotopy | p. 170 |
| The projective factors | p. 174 |
| Operations, turns, and connectivity | p. 174 |
| The Spinor Representations | p. 177 |
| Determination of the projective factors | p. 177 |
| The intertwining theorem | p. 179 |
| The character theorem | p. 182 |
| The irreducible representations | p. 182 |
| The projective factors from the Euler-Rodrigues parameters | p. 183 |
| Inverses and conjugates in the Euler-Rodrigues parametrization | p. 187 |
| Conjugation and the choice of the positive hemisphere | p. 189 |
| The character theorem proved in the Euler-Rodrigues parametrization | p. 190 |
| The SU(2) representation of SO(3) | p. 191 |
| C[subscript i] and the irreducible representations of O(3). The SU'(2) representation of O(3) | p. 194 |
| The representations of C[subscript i] | p. 194 |
| The irreducible representations of O(3) | p. 196 |
| The SU'(2) representation of O(3) | p. 197 |
| The factor system for O(3) | p. 197 |
| Improper point groups | p. 199 |
| The Algebra of Rotations: Quaternions | p. 201 |
| An entertainment on binary rotations | p. 201 |
| The definition of quaternions | p. 202 |
| Inversion of quaternions. Characterization of their scalar and vector parts | p. 205 |
| Conjugate and normalized quaternions. Inverse quaternions | p. 206 |
| The quaternion units | p. 209 |
| SO(3), SU(2), and quaternions | p. 211 |
| Exponential form of quaternions | p. 213 |
| The conical transformation | p. 214 |
| The rectangular transformation | p. 217 |
| Quaternion algebra and the Clifford algebra | p. 219 |
| In praise of mirrors | p. 221 |
| Applications: angle and axis of rotation and SU(2) matrices in terms of Euler angles | p. 223 |
| Double Groups | p. 225 |
| Introduction and example | p. 226 |
| The double group in the quaternion parametrization | p. 230 |
| Notation and operational rules | p. 231 |
| Intertwining | p. 233 |
| Class structure: Opechowski Theorem | p. 235 |
| The Irreducible Representations of SO(3) | p. 237 |
| More about spinor bases | p. 237 |
| The irreducible representation | p. 241 |
| The bases of the representations | p. 246 |
| Examples and Applications | p. 248 |
| The choice of the positive hemisphere | p. 248 |
| Parametrization of the group elements for D[subscript 6], D[subscript 3], C[subscript 3v]. Multiplication tables and factor systems | p. 251 |
| The standard representation | p. 253 |
| The irreducible projective and vector representations | p. 255 |
| The representations of D[subscript 3] | p. 259 |
| The representations of C[subscript 3v] | p. 260 |
| The double group D[subscript 3] | p. 261 |
| Some applications | p. 262 |
| Solutions to Problems | p. 266 |
| References | p. 298 |
| Index | p. 305 |
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