Problems and solutions in group theory for physicists / by Z.Q.Ma and X.Y.Gu
ISBN 981-238-832-X--ISBN981-238-833-8
I. Group Theory. 2. Mathematical Physics. I.Gu, X.Y.(Xiao-Yan) II. Title.
Copyright Ó2004 by World Scientific Publishing Co, Pte. Ltd.
Contents
Preface
1. Review on linear algebras
1.1 Eigenvalues and Eigenvectors of a matrix
1.2 Some special matrices
1.3 Similarity transformation
2. Group and its subsets
2.1 Definition of a group
2.2 Subsets in a group
2.3 Homomorphism of groups
3. Theory of representations
3.1 Transformation operators fro a scalar function
3.2 Inequivalent and irreducible representations
3.3 Subduced and induced representations
3.4 The Clebsch-Gordan coefficients
4. Three-dimensional rotation group
4.1 SO(3) group and its covering group SU(2)
4.2 Inequivalent and irreducible representations
4.3 Lie groups and Lie theorems
4.4 Irreducible tensor operators
4.5 Unitary representations with infinite dimensions
5. Symmetry of crystals
5.1 Symmetric operations and space groups
5.2 Symmetric elements
5.3 International notations for space groups
6. Permutation groups
6.1 Multiplication of permutations
6.2 Young patterns, young tableaux and young operators
6.3 Primitive idempotents in the group algebra
6.4 Irreducible representations and characters
6.5 The inner and outer products of representations
7. Lie groups and Lie algebras
7.1 Classification of semisimple Lie algebras
7.2 Irreducible representations and the Chevalley bases
7.3 Reduction of the direct product of representations
8. Unitary groups
8.1 The SU(N) group and its Lie algebra
8.2 Irreducible tensor representations of SU(N)
8.3 Orthonormal bases for irreducible representations
8.4 Subduced representations
8.5 Casimir invariants of SU(N)
9. Real orthogonal groups
9.1 Tensor representstions of SO(N)
9.2 Spinor representations of SO(N)
9.3 SO(4) group and the Lorentz group
10. The symplectic groups
10.1 The groups sp(2l,R) and usp(2l)
10.2 Irreducible representations of sp(2l)
Bibliography
Index
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