## Group Theory in Physics: Problems and Solutions, Volume 2This solutions booklet is a supplement to the text book 'Group Theory in Physics' by Wu-Ki Tung. It will be useful to lecturers and students taking the subject as detailed solutions are given. |

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### Contents

Basic Group Theory | 1 |

Group Representations | 11 |

General Properties of Irreducible Vectors and Operators | 27 |

Representatons of the Symmetric Groups | 35 |

OneDimensional Continuous Groups | 43 |

Rotations in 3Dimensional Space The Group SO3 | 45 |

The Group SU2 and More About SO3 | 61 |

Euclidean Groups in Two and ThreeDimensional Space | 73 |

The Lorentz and Poincare Groups and SpaceTime Symmetries | 83 |

Space Inversion Invariance | 97 |

Time Reversal Invariance | 103 |

FiniteDimensional Representations of the Classical Groups | 107 |

### Common terms and phrases

2-dimensional representation action amplitudes angular momentum basis basis tensors canonical basis Chapter character table Clebsch-Gordan coefficients complex conjugation components defined by Eq definition denote Derive diagonal dihedral group dimensional dimensions direct product eigenstates eigenvalues eigenvectors equivalence classes explicitly expression factor group follows from Eq given by Eq GL(m Group Theory helicity hermitian identity implies indices inequivalent integration invariant subgroups invariant subspace invariant tensor irreducible representations isomorphic label Lie algebra linear combination Lorentz group Lorentz transformations matrix element matrix multiplication matrix realization Multiply both sides notation Observe obtained in Problem orthogonality orthonormality permutations previous problem Problem 3.1 projection operators Prove rank-2 anti-symmetric tensors recursion relations Rename representation matrices result rewrite rotations satisfy scalar second rank tensor Similarly SOLUTION space inversion space of rank-2 span spherical harmonics step follows subgroups of order symmetric tensors tensor space Theorem trivial unitary vector space