Problems & Solutions in Group Theory for Physicists

PROBLEMS &

SOLUTIONS IN

GROUP TH€ORY

FOR PHYSICISTS

This page intentionally left blank

PROBLEMS &

SOLUTIONS IN

GROUP THEORY

FOR PHYSICISTS

Zhong-Qi

Xicro-Yan

MQ

Gu

Institute of High Energy Physics

China

vp World Scientific

NEW JERSEY - LONDON * SINGAPORE BElJlNG * SHANGHAI - HONG KONG TAIPEI * CHENNAI

Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA oflce: Suite 202, 1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data

Ma, Zhongqi, 1940 -

p. cm.

Problems and solutions in group theory for physicists / by Z.Q. Ma and X.Y. Gu.

Includes bibliographical references and index.

ISBN 98 1-238-832-X (alk. paper) -- ISBN 98 1-238-833-8 (pbk.: alk. paper)

1. Group theory. 2. Mathematical physics. I. Gu, X.Y. (Xiao-Yan). 11. Title.

QC20.7.G76 M3 2004

530.15’22--d~22 2004041980

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library.

Copyright 0 2004 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means,

electronic or mechanical, including photocopying, recording or any informution storage und retrievul

system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright

Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to

photocopy is not required from the publisher.

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface

Group theory is a powerful tool for studying the symmetry of a physical

system, especially the symmetry of a quantum system. Since the exact

solution of the dynamic equation in the quantum theory is generally difficult

to obtain, one has to find other methods to analyze the property of the

system. Group theory provides an effective method by analyzing symmetry

of the system to obtain some precise information of the system verifiable

with observations. Now, Group Theory is a required course for graduate

students major in physics and theoretical chemistry.

The course of Group Theory for the students major in physics is very

different from the same course for those major in mathematics. A graduate

student in physics needs to know the theoretical framework of group theory

and more importantly to master the techniques in application of group

theory to various fields of physics, which is actually his main objective for

taking the class. However, no course or textbook on group theory can be

expected to include all explicit solutions to every problem of group theory

in his research field of physics. A student of physics has to know the

fundamental theory of group theory, otherwise he may not be able to apply

the techniques creatively. On the other hand, physics students are not

expected to completely grasp all the mathematics behind group theory due

to the breadth of the knowledge required.

One of the authors (Ma) first taught the group theory course in 1962.

Since 1986, he has been teaching Group Theory to graduate students mainly

major in physics at the Graduate School of Chinese Academy of Sciences.

In addition, most of his research work has been related to applications of

group theory to physics. In 1996 the Chinese Academy of Sciences decided

to publish a series of textbooks for graduate students. He was invited to

write a textbook on group theory for the series. In his book, based on his

v

vi Problems and Solutions in Group Theory

experience in teaching and research work, he explained the fundamental

concepts and techniques of group theory progressively and systematically

using the language familiar to physicists, and also emphasized the ways with

which group theory is applied to physics. The textbook (in Chinese) has

been widely used for Group Theory classes in China since it was published

by Science Press in Beijing six years ago. He is honored and flattered by

the tremendous reception the book has received.

By the request of the readers, an exercise book on group theory by the

same author was published in 2002 by Science Press to form a complete

set of textbooks on group theory. In order to make the exercise book

self-contained, a brief review of the main concepts and techniques is given

before the problems in each section. The reviews can be used as a concise

textbook on group theory. The present book is the new edition of that

book. A great deal of new materials drawn from teaching and research

experiences since the publication of the previous edition are included. The

reviews of each chapter has been extensively revised. Last four chapters

are essentially new.

This book consists of ten chapters. Chapter 1 is a short review on lin￾ear algebra. The reader is required not only to be familiar with its basic

concepts but also to master its applications, especially the similarity trans￾formation method. In Chapter 2, the concepts of a group and its subsets

are studied through examples of some finite groups, where the importance

of the multiplication table of a finite group is emphasized. Readers should

pay special attention to Problem 17 of Chapter 2 and Problem 14 of Chap￾ter 3 which demonstrate a systematic method for analyzing a finite group.

The theory of representations of a group is studied in Chapter 3. The trans￾formation operator PR for the scalar functions bridges the gap between the

representation theory and the physical application. The subduced and in￾duced representations of groups are used to construct the character tables

of finite groups in Chapter 3, and to study the outer product of representa￾tions of the permutation groups in Chapter 6. The symmetry groups T for

a tetrahedron, 0 for a cube and I for an icosahedron are studied in Chap￾ters 3 and 4. The Clebsch-Gordan series and coefficients are introduced

in Chapter 3 and are calculated for various situations in the subsequent

chapters. The calculated results of the Clebsch-Gordan coefficients and

the Clebsch-Gordan series for the group I and for the permutation groups

listed in Problem 27 of Chapter 3 and Problem 31 of Chapter 6, due to its

complexity, are only for reference.

The classification and representations of semisimple Lie algebras are

Preface vi i

introduced in Chapter 7 and partly in Chapter 4 by the language familiar

to physicists. The methods of block weight diagrams and dominant weight

diagrams are recommended for calculating the representation matrices of

the generators and the Clebsch-Gordan series in a simple Lie algebra. The

readers who are interested in the strict mathematical definitions and proofs

in the theory of semisimple Lie algebras are recommended to read the more

mathematically oriented books, e.g. [Bourbaki (1989)l.

The remaining part of the book is devoted to the properties of some

important symmetry groups of physical systems. In Chapter 4 the symme￾try group SO(3) of a spherically symmetric system in three dimensions is

studied. The unitary representations with infinite dimensions of the non￾compact group are discussed with the simplest example SO(2,l) in Problem

25 of Chapter 4. In Chapter 5 we introduce the symmetry of the crystals.

More attention should be paid to the analysis method for the symmetry

of a crystal from its International Notation (Problem 6). The commonly

used matrix groups are studied in the last three chapters, while the Lorentz

group is briefly discussed in Chapter 9.

The systematic examination of Young operators is an important char￾acteristic of this book. We calculate the characters, the representation ma￾trices, and the outer product of the irreducible representations of the per￾mutation groups using the Young operators. The method of block weight

diagrams can only symbolically give the basis states in the representation

space of a Lie algebra. However, for the matrix groups SU(N), SO(N)

and Sp(24, which are related to four classical Lie algebras, the basis states

can be explicitly calculated using the Young operators. The relationship

between two methods for the irreducible representations of the classical

Lie algebras is demonstrated in the last three chapters in detail. The di￾mensions of the irreducible representations of the permutation groups, the

SU(N) groups, the SO(N) groups, and the Sp(2l) groups are all calculated

with the hook rule, a method based on the Young diagram.

In summary, this book is written mainly for physics students and young

physicists. Great, care has been taken to make the book as self-contained as

possible. However, this book reflects mainly the experiences of the authors.

We sincerely welcome any suggestions and comments from the readers. This

book was supported by the National Natural Science Foundation of China.

Institute of High Energy Physics

Beijing, China

December, 2003

Zhong-Qi Ma

Xiao-Yan Gu

This page intentionally left blank

Contents

Preface V

1 . REVIEW ON LINEAR ALGEBRAS 1

1.1 Eigenvalues and Eigenvectors of a Matrix .......... 1

1.2 Some Special Matrices ..................... 1.3 Similarity Transformation ................... 7

4

2 . GROUP AND ITS SUBSETS 27

29

33

2.1 Definition of a Group ...................... 27

Subsets in a Group .......................

2.3 Homomorphism of Groups ...................

2.2

3 . THEORY OF REPRESENTATIONS 43

3.1 Transformation Operators for a Scalar.Function ....... 43

3.2 Inequivalent and Irreducible Representations ........ 47

3.3 Subduced and Induced Representations ........... 65

3.4 The Clebsch-Gordan Coefficients ............... 79

4 . THREE-DIMENSIONAL ROTATION GROUP 115

4.1 SO(3) Group and Its Covering Group SU(2) ......... 115

4.2 Inequivalent and Irreducible Representations ........ 123

Lie Groups and Lie Theorems ................. 140

4.4 Irreducible Tensor Operators ................. 146

4.5 Unitary Representations with Infinite Dimensions ...... 166

4.3

5 . SYMMETRY OF CRYSTALS 173

5.1 Symmetric Operations and Space Groups .......... 173

ix

X Problems and Solutions in Group Theory

5.2 Symmetric Elements ......................

5.3 International Notations for Space Groups ..........

6 . PERMUTATION GROUPS

6.1 Multiplication of Permutations ................

6.4 Irreducible Representations and Characters

6.2 Young Patterns, Young Tableaux and Young Operators . .

6.3 Primitive Idempotents in the Group Algebra ........

6.5 The Inner and Outer Products of Representations .....

.........

7 . LIE GROUPS AND LIE ALGEBRAS

7.2 Irreducible Representations and the Chevalley Bases ....

7.1 Classification of Semisimple Lie Algebras ..........

7.3 Reduction of the Direct Product of Representations ....

8 . UNITARY GROUPS

8.1

8.2 Irreducible Tensor Representations of SU(N) ........

8.3 Orthonormal Bases for Irreducible Representations .....

The SU(N) Group and Its Lie Algebra ............

8.4 Subduced Representations ...................

8.5 Casimir Invariants of SU(N) .............. .. ..

9 . REAL ORTHOGONAL GROUPS

9.1

9.3

Tensor Represent ations of SO (N) ............... 9.2 Spinor Representations of SO(N) ...............

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( 2t) .............

177

186

193

193

197

205

211

237

269

269

279

299

317

317

321

336

362

369

375

375

403

415

433

433

440

Bibliography 457

Index 461

PROBKMS &I

SOLUTIONS IN

GROUP THEORY

FOR PHYSICISTS

This page intentionally left blank

Chapter 1

REVIEW ON LINEAR ALGEBRAS

1.1 Eigenvalues and Eigenvectors of a Matrix

* The eigenequation of a matrix R is

Ra = Xa, (1.1)

where X is the eigenvalue and a is the eigenvector for the eigenvalue. An

eigenvector is determined up to a constant factor. A null vector is a trivial

eigenvector of any matrix. We only discuss nontrivial eigenvectors.

Equation (1.1) is a set of linear homogeneous equations with respect to

the components ap. The necessary and sufficient condition for the existence

of a nontrivial solution to Eq. (1.1) is the vanishing of the coefficient

determinant :

= + (-X)”-’TrR + , . . + det R = 0,

where rn is the dimension of the matrix R, TrR is its trace (the sum of the

diagonal elements), and detR denotes its determinant. Equation (1.2) is

called the secular equation of R. Evidently, this is an algebraic equation

of order m with respect to A, and there are rn complex roots including

multiple roots. Each root is an eigenvalue of the matrix. For a given

eigenvalue A, there is at least one eigenvector a obtained from solving Eq.

(1.1). However, for a root X with multiplicity n, it is not certain to obtain

n linearly independent eigenvectors from Eq. (1.1).

1

2 Problems and Solutions in Group Theory

The rank of a matrix R is said to be r if only r vectors among rn

row-vectors (or column-vectors) of R are linearly independent. If the rank

of (R - Xl) is r, then (rn - r) linearly independent eigenvectors can be

obtained from Eq. (1.1). Any linear combination of eigenvectors for a

given eigenvalue is still an eigenvector for this eigenvalue.

For a lower-dimensional matrix or for a sparse matrix (with many zero

matrix entries), its eigenvalues and eigenvectors can be obtained by guess

from experience or from some known results. So long as the eigenvalue and

the eigenvector satisfy the eigenequation (l.l), they are the correct results.

On the other hand, even if the results are calculated, they should also be

checked whether Eq. (1.1) is satisfied. * A matrix R is said to be hermitian if Rt = R. R is said to be unitary

if Rt = R-l. A real and hermitian matrix is a real symmetric matrix. A

real and unitary matrix is a real orthogonal matrix. R is said to be positive

definite if its eigenvalues are all positive. R is said to be positive semi￾definite if its eigenvalues are non-negative. A negative definite or negative

semidefinite matrix can be defined similarly.

1. Prove that the sum of the eigenvalues of a matrix is equal to the trace of

the matrix, and the product of eigenvalues is equal to the determinant

of the matrix.

Solution. The secular equation of an m-dimensional matrix R is an alge￾braic equation of order rn. Its rn complex roots, including multiple roots,

are the eigenvalues of the matrix R. So, the secular equation can also be

expressed in the following way:

m

det(R- Xl) = n (Xj -A)

j=l m m

j= 1 j=1

= 0.

Comparing it with Eq. (1.2), we obtain that the sum of the eigenvalues is

the trace of the matrix and the product of the eigenvalues is equal to the

determinant of the matrix.

Review on Linear Algebras 3

2. Calculate the eigenvalues and eigenvectors of the Pauli matrices 01 and

u2 :

a,=(; ;), 02=(i 0 -i 0).

Solution. The action of 01 on a vector is to interchange two components

of the vector. A vector is an eigenvector of 01 for the eigenvalue 1 if its two

components are equal. A vector is an eigenvector of 01 for the eigenvalue

-1 if its two components are different by sign.

.1(:)=(:>. Q(!l)= - (1').

Similarly, if two components of a vector are different by a factor fi, then

it is an eigenvector of 02:

.2(])=(]), ",( -i '>= - ('). -i

Sometimes a matrix R may contain a submatrix 01 or 02 in the form of

direct sum or direct product. Thus, some eigenvalues and eigenvectors of

R can be obtained in terms of the above results. A matrix R is said to

contain a two-dimensional submatrix in the form of direct sum if for given

a and b, R,, = R,, = Rbc = Rcb = 0, where c is not equal to a and b.

3. Calculate the eigenvalues and eigenvectors of the matrix R

Solution. R can be regarded as the direct sum of two submatrices 01,

one lies in the first and fourth rows (columns), the other in the second and

third rows (columns). F'rom the result of Problem 2, two eigenvalues of R

are 1, the remaining two are -1. The relative eigenvectors are as follows.

1: (i), (i). -l: (i), -1 (;l).