GroupheoryM.S. Dresselhaus

    G. Dresselhaus

    A. Jorio

    Group Theory

    Application to the Physics of Condensed Matter

    With 131 Figures and 219 Tables

    123Professor Dr. Mildred S. Dresselhaus

    Dr. Gene Dresselhaus

    Massachusetts Institute of Technology Room 13-3005

    Cambridge, MA, USA

    E-mail: millie@mgm.mit.edu, gene@mgm.mit.edu

    Professor Dr. Ado Jorio

    Departamento de Física

    Universidade Federal de Minas Gerais

    CP702 – Campus, Pampulha

    Belo Horizonte,MG, Brazil 30.123-970

    E-mail: adojorio@?sica.ufmg.br

    ISBN 978-3-540-32897-1 e-ISBN 978-3-540-32899-8

    DOI 10.1007978-3-540-32899-8

    Library of Congress Control Number: 2007922729

    2008 Springer-Verlag Berlin Heidelberg

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    springer.comThe authors dedicate this book

    to John Van Vleck and Charles KittelPreface

    Symmetry can be seen as the most basic and important concept in physics.

    Momentum conservation is a consequence of translational symmetry of space.

    More generally, every process in physics is governed by selection rules that

    are the consequence of symmetry requirements. On a given physical system,the eigenstate properties and the degeneracy of eigenvalues are governed by

    symmetry considerations. The beauty and strength of group theory applied to

    physics resides in the transformation of many complex symmetry operations

    into a very simple linear algebra. The concept of representation, connecting

    the symmetry aspects to matrices and basis functions, together with a few

    simple theorems, leads to the determination and understanding of the funda-

    mental properties of the physical system, and any kind of physical property,its transformations due to interactions or phase transitions, are described in

    terms of the simple concept of symmetry changes.

    The reader may feel encouraged when we say group theory is “simple linear

    algebra.” It is true that group theory may look complex when either the math-

    ematical aspects are presented with no clear and direct correlation to applica-

    tions in physics, or when the applications are made with no clear presentation

    of the background. The contact with group theory in these terms usually leads

    to frustration, and although the reader can understand the speci?c treatment,he (she) is unable to apply the knowledge to other systems of interest. What

    this book is about is teaching group theory in close connection to applications,so that students can learn, understand, and use it for their own needs.

    This book is divided into six main parts. Part I, Chaps. 1–4, introduces

    the basic mathematical concepts important for working with group theory.

    Part II, Chaps. 5 and 6, introduces the ?rst application of group theory to

    quantum systems, considering the e?ect of a crystalline potential on the elec-

    tronic states of an impurity atom and general selection rules. Part III, Chaps. 7

    and 8, brings the application of group theory to the treatment of electronic

    states and vibrational modes of molecules. Here one ?nds the important group

    theory concepts of equivalence and atomic site symmetry. Part IV, Chaps. 9

    and 10, brings the application of group theory to describe periodic lattices in

    both real and reciprocal lattices. Translational symmetry gives rise to a lin-

    ear momentum quantum number and makes the group very large. Here theVIII Preface

    concepts of cosets and factor groups, introduced in Chap. 1, are used to factor

    out the e?ect of the very large translational group, leading to a ?nite group

    to work with each unique type of wave vector – the group of the wave vector.

    Part V, Chaps. 11–15, discusses phonons and electrons in solid-state physics,considering general positions and speci?c high symmetry points in the Bril-

    louin zones, and including the addition of spins that have a 4π rotation as the

    identity transformation. Cubic and hexagonal systems are used as general ex-

    amples. Finally, Part VI, Chaps. 16–18, discusses other important symmetries,such as time reversal symmetry, important for magnetic systems, permutation

    groups, important for many-body systems, and symmetry of tensors, impor-

    tant for other physical properties, such as conductivity, elasticity, etc.

    This book on the application of Group Theory to Solid-State Physics grew

    out of a course taught to Electrical Engineering and Physics graduate students

    by the authors and developed over the years to address their professional

    needs. The material for this book originated from group theory courses taught

    by Charles Kittel at U.C. Berkeley and by J.H. Van Vleck at Harvard in the

    early 1950s and taken by G. Dresselhaus and M.S. Dresselhaus, respectively.

    The material in the book was also stimulated by the classic paper of Bouckaert,Smoluchowski, and Wigner [1], which ?rst demonstrated the power of group

    theory in condensed matter physics. The diversity of applications of group

    theory to solid state physics was stimulated by the research interests of the

    authors and the many students who studied this subject matter with the

    authors of this volume. Although many excellent books have been published

    on this subject over the years, our students found the speci?c subject matter,the pedagogic approach, and the problem sets given in the course user friendly

    and urged the authors to make the course content more broadly available.

    The presentation and development of material in the book has been tai-

    lored pedagogically to the students taking this course for over three decades

    at MIT in Cambridge, MA, USA, and for three years at the University Fed-

    eral of Minas Gerais (UFMG) in Belo Horizonte, Brazil. Feedback came from

    students in the classroom, teaching assistants, and students using the class

    notes in their doctoral research work or professionally.

    We are indebted to the inputs and encouragement of former and present

    students and collaborators including, Peter Asbeck, Mike Kim, Roosevelt Peo-

    ples, Peter Eklund, Riichiro Saito, Georgii Samsonidze, Jose Francisco de Sam-

    paio, Luiz Gustavo Can? cado, and Eduardo Barros among others. The prepa-

    ration of the material for this book was aided by Sharon Cooper on the ?gures,Mario Hofmann on the indexing and by Adelheid Duhm of Springer on editing

    the text. The MIT authors of this book would like to acknowledge the contin-

    ued long term support of the Division of Materials Research section of the US

    National Science Foundation most recently under NSF Grant DMR-04-05538.

    Cambridge, Massachusetts USA, Mildred S. Dresselhaus

    Belo Horizonte, Minas Gerais, Brazil, Gene Dresselhaus

    August 2007 Ado JorioContents

    Part I Basic Mathematics

    1 Basic Mathematical Background: Introduction ............. 3

    1.1 De?nitionofaGroup.................................... 3

    1.2 SimpleExampleofaGroup .............................. 3

    1.3 BasicDe?nitions........................................ 6

    1.4 RearrangementTheorem................................. 7

    1.5 Cosets................................................. 7

    1.6 ConjugationandClass................................... 9

    1.7 FactorGroups.......................................... 11

    1.8 GroupTheoryandQuantumMechanics ................... 11

    2 Representation Theory and Basic Theorems ............... 15

    2.1 ImportantDe?nitions ................................... 15

    2.2 Matrices............................................... 16

    2.3 IrreducibleRepresentations .............................. 17

    2.4 TheUnitarityofRepresentations ......................... 19

    2.5 Schur’sLemma(Part1) ................................. 21

    2.6 Schur’sLemma(Part2) ................................. 23

    2.7 Wonderful Orthogonality Theorem . . . . . . 25

    2.8 RepresentationsandVectorSpaces........................ 28

    3 Character of a Representation ............................. 29

    3.1 De?nitionofCharacter .................................. 29

    3.2 CharactersandClass.................................... 30

    3.3 Wonderful Orthogonality Theorem for Character . . . 31

    3.4 ReducibleRepresentations ............................... 33

    3.5 TheNumberof IrreducibleRepresentations ................ 35

    3.6 Second Orthogonality Relation for Characters . . . . 36

    3.7 RegularRepresentation.................................. 37

    3.8 SettingupCharacterTables.............................. 40XContents

    3.9 Schoen?iesSymmetryNotation........................... 44

    3.10 TheHermann–MauguinSymmetryNotation ............... 46

    3.11 Symmetry Relations and Point Group Classi?cations . . 48

    4 Basis Functions ............................................ 57

    4.1 SymmetryOperationsandBasisFunctions................. 57

    4.2 BasisFunctions for IrreducibleRepresentations ............. 58

    4.3 Projection Operators ? P(Γn)

    kl.............................. 64

    4.4 Derivation of an Explicit Expression for ? P(Γn)

    k .............. 64

    4.5 The E?ect of Projection Operations on an Arbitrary Function 65

    4.6 Linear Combinations of Atomic Orbitals for Three

    Equivalent Atoms at the Corners of an Equilateral Triangle . . 67

    4.7 The Application of Group Theory to Quantum Mechanics . . 70

    Part II Introductory Application to Quantum Systems

    5 Splitting of Atomic Orbitals in a Crystal Potential ......... 79

    5.1 Introduction ........................................... 79

    5.2 Charactersfor theFullRotationGroup.................... 81

    5.3 Cubic Crystal Field Environment

    foraParamagneticTransitionMetal Ion................... 85

    5.4 CommentsonBasisFunctions ............................ 90

    5.5 CommentsontheFormofCrystalFields .................. 92

    6 Application to Selection Rules and Direct Products ....... 97

    6.1 TheElectromagneticInteractionasaPerturbation.......... 97

    6.2 Orthogonality of Basis Functions . . . . . . 99

    6.3 DirectProductofTwoGroups ...........................100

    6.4 DirectProductofTwoIrreducibleRepresentations..........101

    6.5 Charactersfor theDirectProduct.........................103

    6.6 Selection Rule Concept in Group TheoreticalTerms.........105

    6.7 ExampleofSelectionRules...............................106

    Part III Molecular Systems

    7 Electronic States of Molecules and Directed Valence ....... 113

    7.1 Introduction ...........................................113

    7.2 GeneralConceptofEquivalence ..........................115

    7.3 DirectedValenceBonding................................117

    7.4 DiatomicMolecules .....................................118

    7.4.1 HomonuclearDiatomicMolecules ...................118

    7.4.2 HeterogeneousDiatomicMolecules..................120Contents XI

    7.5 ElectronicOrbitals forMultiatomicMolecules ..............124

    7.5.1 The NH3 Molecule................................124

    7.5.2 The CH4 Molecule ................................125

    7.5.3 The Hypothetical SH6 Molecule ....................129

    7.5.4 The Octahedral SF6 Molecule ......................133

    7.6 σ-and π-Bonds.........................................134

    7.7 Jahn–TellerE?ect ......................................141

    8 Molecular Vibrations, Infrared, and Raman Activity....... 147

    8.1 MolecularVibrations:Background ........................147

    8.2 Application of Group Theory to Molecular Vibrations . . 149

    8.3 FindingtheVibrationalNormalModes ....................152

    8.4 Molecular Vibrations in H2O.............................154

    8.5 OvertonesandCombinationModes .......................156

    8.6 InfraredActivity........................................157

    8.7 RamanE?ect ..........................................159

    8.8 Vibrations forSpeci?cMolecules..........................161

    8.8.1 TheLinearMolecules .............................161

    8.8.2 Vibrations of the NH3 Molecule ....................166

    8.8.3 Vibrations of the CH4 Molecule ....................168

    8.9 RotationalEnergyLevels ................................170

    8.9.1 TheRigidRotator ................................170

    8.9.2 Wigner–EckartTheorem...........................172

    8.9.3 Vibrational–Rotational Interaction . . . . 174

    Part IV Application to Periodic Lattices

    9 Space Groups in Real Space ............................... 183

    9.1 MathematicalBackgroundforSpaceGroups ...............184

    9.1.1 SpaceGroupsSymmetryOperations ................184

    9.1.2 CompoundSpaceGroupOperations ................186

    9.1.3 Translation Subgroup . . . . . . . 188

    9.1.4 Symmorphic and Nonsymmorphic Space Groups . . 189

    9.2 BravaisLatticesandSpaceGroups........................190

    9.2.1 ExamplesofSymmorphicSpaceGroups .............192

    9.2.2 Cubic Space Groups

    andtheEquivalenceTransformation ................194

    9.2.3 ExamplesofNonsymmorphicSpaceGroups ..........196

    9.3 Two-DimensionalSpaceGroups ..........................198

    9.3.1 2DObliqueSpaceGroups..........................200

    9.3.2 2DRectangularSpaceGroups......................201

    9.3.3 2DSquareSpaceGroup ...........................203

    9.3.4 2D Hexagonal Space Groups . . . . . . 203

    9.4 LineGroups............................................204XII Contents

    9.5 The Determination of Crystal Structure and Space Group . . 205

    9.5.1 Determinationof theCrystalStructure ..............206

    9.5.2 Determinationof theSpaceGroup ..................206

    10 Space Groups in Reciprocal Space and Representations .... 209

    10.1 ReciprocalSpace........................................210

    10.2 Translation Subgroup . . . . . . . . 211

    10.2.1 Representations for the Translation Group . . . 211

    10.2.2 Bloch’s Theorem and the Basis

    oftheTranslationalGroup.........................212

    10.3 Symmetry of k Vectors and the Group of the Wave Vector . . 214

    10.3.1 Point Group Operation in r-space and k-space .......214

    10.3.2 The Group of the Wave Vector Gk and the Star of k . . 215

    10.3.3 E?ect of Translations and Point Group Operations

    onBlochFunctions ...............................215

    10.4 SpaceGroupRepresentations.............................219

    10.4.1 SymmorphicGroupRepresentations.................219

    10.4.2 Nonsymmorphic Group Representations

    andtheMultiplierAlgebra.........................220

    10.5 Characters for the EquivalenceRepresentation..............221

    10.6 Common Cubic Lattices: Symmorphic Space Groups . . 222

    10.6.1 The Γ Point .....................................223

    10.6.2 Points with k 

    =0.................................224

    10.7 Compatibility Relations . . . . . . . 227

    10.8 The Diamond Structure: Nonsymmorphic Space Group . . 230

    10.8.1 Factor Group and the Γ Point......................231

    10.8.2 Points with k 

    =0.................................232

    10.9 Finding Character Tables for all Groups of the Wave Vector . . 235

    Part V Electron and Phonon Dispersion Relation

    11 Applications to Lattice Vibrations ......................... 241

    11.1 Introduction ...........................................241

    11.2 LatticeModesandMolecularVibrations...................244

    11.3 ZoneCenterPhononModes ..............................246

    11.3.1 TheNaClStructure...............................246

    11.3.2 ThePerovskiteStructure ..........................247

    11.3.3 Phonons in the Nonsymmorphic Diamond Lattice . . 250

    11.3.4 PhononsintheZincBlendeStructure ...............252

    11.4 Lattice Modes Away from k =0 ..........................253

    11.4.1 Phonons in NaCl at the X Point k =(πa)(100) . . 254

    11.4.2 Phonons in BaTiO3 at the X Point .................256

    11.4.3 Phonons at the K Point in Two-Dimensional Graphite . 258Contents XIII

    11.5 Phonons in Te and α-Quartz Nonsymmorphic Structures . . 262

    11.5.1 PhononsinTellurium .............................262

    11.5.2 Phonons in the α-QuartzStructure .................268

    11.6 E?ectofAxialStressonPhonons .........................272

    12 Electronic Energy Levels in a Cubic Crystals .............. 279

    12.1 Introduction ...........................................279

    12.2 Plane Wave Solutions at k =0 ...........................282

    12.3 Symmetrized Plane Solution Waves along the Δ-Axis........286

    12.4 Plane Wave Solutions at the X Point......................288

    12.5 E?ectofGlidePlanesandScrewAxes.....................294

    13 Energy Band Models Based on Symmetry ................. 305

    13.1 Introduction ...........................................305

    13.2 k · p PerturbationTheory................................307

    13.3 Example of k · p Perturbation Theory

    for a Nondegenerate Γ+

    1 Band............................308

    13.4 Two Band Model:

    DegenerateFirst-OrderPerturbationTheory ...............311

    13.5 Degenerate second-order k · p PerturbationTheory..........316

    13.6 Nondegenerate k · p Perturbation Theory at a Δ Point ......324

    13.7 Use of k · p Perturbation Theory

    toInterpretOpticalExperiments .........................326

    13.8 Application of Group Theory to Valley–Orbit Interactions

    in Semiconductors . . . . . . . . 327

    13.8.1 Background......................................328

    13.8.2 Impurities in Multivalley Semiconductors . . . 330

    13.8.3 TheValley–OrbitInteraction.......................331

    14 Spin–Orbit Interaction in Solids and Double Groups ....... 337

    14.1 Introduction ...........................................337

    14.2 CrystalDoubleGroups ..................................341

    14.3 DoubleGroupProperties ................................343

    14.4 Crystal Field Splitting Including Spin–Orbit Coupling . . 349

    14.5 Basis Functions for Double Group Representations . . . 353

    14.6 SomeExplicitBasisFunctions............................355

    14.7 Basis Functions for Other Γ+

    8 States ......................358

    14.8 Comments on Double Group Character Tables . . . . 359

    14.9 Plane Wave Basis Functions

    forDoubleGroupRepresentations ........................360

    14.10 Group of the Wave Vector

    forNonsymmorphicDoubleGroups .......................362XIV Contents

    15 Application of Double Groups to Energy Bands with Spin . 367

    15.1 Introduction ...........................................367

    15.2 E(k) for the Empty Lattice Including Spin–Orbit Interaction . 368

    15.3 The k · p Perturbation with Spin–Orbit Interaction . . 369

    15.4 E(k) for a Nondegenerate Band Including

    Spin–OrbitInteraction ..................................372

    15.5 E(k) for Degenerate Bands Including Spin–Orbit Interaction . 374

    15.6 E?ective g-Factor.......................................378

    15.7 Fourier Expansion of Energy Bands: Slater–Koster Method . . 389

    15.7.1 Contributions at d =0 ............................396

    15.7.2 Contributions at d =1 ............................396

    15.7.3 Contributions at d =2 ............................397

    15.7.4 Summing Contributions through d =2 ..............397

    15.7.5 OtherDegenerateLevels...........................397

    Part VI Other Symmetries

    16 Time Reversal Symmetry .................................. 403

    16.1 TheTimeReversalOperator .............................403

    16.2 PropertiesoftheTimeReversalOperator..................404

    16.3 The E?ect of ? T on E(k),NeglectingSpin ..................407

    16.4 The E?ect of ? T on E(k), Including

    theSpin–Orbit Interaction ...............................411

    16.5 MagneticGroups .......................................416

    16.5.1 Introduction .....................................418

    16.5.2 TypesofElements ................................418

    16.5.3 TypesofMagneticPointGroups....................419

    16.5.4 Properties of the 58 Magnetic Point Groups {Ai,Mk} . 419

    16.5.5 ExamplesofMagneticStructures ...................423

    16.5.6 E?ect of Symmetry on the Spin Hamiltonian

    for the32OrdinaryPointGroups...................426

    17 Permutation Groups and Many-Electron States ............ 431

    17.1 Introduction ...........................................432

    17.2 Classes and Irreducible Representations

    ofPermutationGroups ..................................434

    17.3 BasisFunctionsofPermutationGroups....................437

    17.4 PauliPrincipleinAtomicSpectra.........................440

    17.4.1 Two-ElectronStates ..............................440

    17.4.2 Three-ElectronStates .............................443

    17.4.3 Four-ElectronStates ..............................448

    17.4.4 Five-ElectronStates...............................451

    17.4.5 General Comments on Many-Electron States . . 451Contents XV

    18 Symmetry Properties of Tensors ........................... 455

    18.1 Introduction ...........................................455

    18.2 Independent Components of Tensors

    UnderPermutationGroupSymmetry......................458

    18.3 Independent Components of Tensors:

    PointSymmetryGroups .................................462

    18.4 Independent Components of Tensors

    UnderFullRotationalSymmetry .........................463

    18.5 TensorsinNonlinearOptics..............................463

    18.5.1 Cubic Symmetry: Oh ..............................464

    18.5.2 Tetrahedral Symmetry: Td .........................466

    18.5.3 Hexagonal Symmetry: D6h .........................466

    18.6 ElasticModulusTensor..................................467

    18.6.1 Full Rotational Symmetry: 3D Isotropy . . . . 469

    18.6.2 IcosahedralSymmetry.............................472

    18.6.3 CubicSymmetry..................................472

    18.6.4 OtherSymmetryGroups ..........................474

    A Point Group Character Tables ............................. 479

    B Two-Dimensional Space Groups ........................... 489

    C Tables for 3D Space Groups ............................... 499

    C.1 RealSpace.............................................499

    C.2 ReciprocalSpace........................................503

    D Tables for Double Groups ................................. 521

    E Group Theory Aspects of Carbon Nanotubes .............. 533

    E.1 Nanotube Geometry and the (n,m) Indices ................534

    E.2 LatticeVectors inRealSpace.............................534

    E.3 LatticeVectors inReciprocalSpace .......................535

    E.4 CompoundOperationsandTubeHelicity ..................536

    E.5 CharacterTables forCarbonNanotubes ...................538

    F Permutation Group Character Tables ...................... 543

    References ..................................................... 549

    Index .......................................................... 553Part I

    Basic Mathematics1

    Basic Mathematical Background: Introduction

    In this chapter we introduce the mathematical de?nitions and concepts that

    are basic to group theory and to the classi?cation of symmetry proper-

    ties [2].

    1.1 De?nition of a Group

    A collection of elements A,B,C,... form a group when the following four

    conditions are satis?ed:

    1. The product of any two elements of the group is itself an element of

    the group. For example, relations of the type AB = C are valid for all

    members of the group.

    2. The associative law is valid – i.e., (AB)C = A(BC).

    3. There exists a unit element E (also called the identity element) such that

    the product of E with any group element leaves that element unchanged

    AE = EA = A.

    4. For every element A there exists an inverse element A?1 such that A?1A =

    AA?1 = E.

    In general, the elements of a group will not commute, i.e., AB 

    = BA.Butif

    all elements of a group commute, the group is then called an Abelian group.

    1.2 Simple Example of a Group

    As a simple example of a group, consider the permutation group for three

    numbers, P(3). Equation (1.1) lists the 3! = 6 possible permutations that

    can be carried out; the top row denotes the initial arrangement of the three

    numbers and the bottom row denotes the ?nal arrangement. Each permutation

    is an element of P(3).4 1 Basic Mathematical Background: Introduction

    Fig. 1.1. The symmetry operations on an equilateral triangle are the rotations by

    ±2π3 about the origin and the rotations by π about the three twofold axes. Here

    the axes or points of the equilateral triangle are denoted by numbers in circles

    E =

    

    123

    123

    

    A =

    

    123

    132

    

    B =

    

    123

    321

    

    C =

    

    123

    213

    

    D =

    

    123

    312

    

    F =

    

    123

    231

    

    . (1.1)

    We can also think of the elements in (1.1) in terms of the three points of an

    equilateral triangle (see Fig. 1.1). Again, the top row denotes the initial state

    and the bottom row denotes the ?nal position of each number. For example,in symmetry operation D, 1 moves to position 2, and 2 moves to position 3,while 3 moves to position 1, which represents a clockwise rotation of 2π3

    (see caption to Fig. 1.1). As the e?ect of the six distinct symmetry operations

    that can be performed on these three points (see caption to Fig. 1.1). We can

    call each symmetry operation an element of the group. The P(3) group is,therefore, identical with the group for the symmetry operations on a equilat-

    eral triangle shown in Fig. 1.1. Similarly, F is a counter-clockwise rotation of

    2π3, so that the numbers inside the circles in Fig. 1.1 move exactly as de?ned

    by Eq. 1.1.

    It is convenient to classify the products of group elements. We write these

    products using a multiplication table. In Table 1.1 a multiplication table is

    written out for the symmetry operations on an equilateral triangle or equiva-

    lently for the permutation group of three elements. It can easily be shown that

    the symmetry operations given in (1.1) satisfy the four conditions in Sect. 1.1

    and therefore form a group. We illustrate the use of the notation in Table 1.1

    by verifying the associative law (AB)C = A(BC) for a few elements:

    (AB)C = DC = B

    A(BC)= AD = B. (1.2)

    Each element of the permutation group P(3) has a one-to-one correspondence

    to the symmetry operations of an equilateral triangle and we therefore say

    that these two groups are isomorphic to each other. We furthermore can1.2 Simple Example of a Group 5

    Table 1.1. Multiplicationa

    table for permutation group of three elements; P(3)

    EABCDF

    E EABCDF

    A AEDFBC

    B BFEDCA

    C CDFEAB

    D DCABFE

    F FBCAED

    a

    AD = B de?nes use of multiplication table

    use identical group theoretical procedures in dealing with physical problems

    associated with either of these groups, even though the two groups arise from

    totally di?erent physical situations. It is this generality that makes group

    theory so useful as a general way to classify symmetry operations arising in

    physical problems.

    Often, when we deal with symmetry operations in a crystal, the geomet-

    rical visualization of repeated operations becomes di?cult. Group theory is

    designed to help with this problem. Suppose that the symmetry operations in

    practical problems are elements of a group; this is generally the case. Then if

    we can associate each element with a matrix that obeys the same multiplica-

    tion table as the elements themselves, that is, if the elements obey AB = D,then the matrices representing the elements must obey

    M(A) M(B)= M(D) . (1.3)

    If this relation is satis?ed, then we can carry out all geometrical opera-

    tions analytically in terms of arithmetic operations on matrices, which are

    usually easier to perform. The one-to-one identi?cation of a generalized sym-

    metry operation with a matrix is the basic idea of a representation and

    why group theory plays such an important role in the solution of practical

    problems.

    A set of matrices that satisfy the multiplication table (Table 1.1) for the

    group P(3) are:

    E =

    

    10

    01

    

    A =

    

    10

    01

    

    B =

     1

    2 ?

    √3

    2

    √3

    2 ?1

    2

    

    C =

     1

    2

    √3

    2 √3

    2 ?1

    2

    

    D =

    

    1

    2

    √3

    2

    √3

    2 ?1

    2

    

    F =

    

    1

    2 ?

    √3

    2 √3

    2 ?1

    2

    

    . (1.4)

    We note that the matrix corresponding to the identity operation E is always

    a unit matrix. The matrices in (1.4) constitute a matrix representation of

    the group that is isomorphic to P(3) and to the symmetry operations on6 1 Basic Mathematical Background: Introduction

    an equilateral triangle. The A matrix represents a rotation by ±π about the

    y axis, while the B and C matrices, respectively, represent rotations by ±π

    about axes 2 and 3 in Fig. 1.1. D and F, respectively, represent rotation of

    2π3and+2π3 around the center of the triangle.

    1.3 Basic De?nitions

    De?nition 1. The order of a group ≡ the number of elements in the group.

    We will be mainly concerned with ?nite groups. As an example, P(3) is of

    order 6.

    De?nition 2. Asubgroup ≡ a collection of elements within a group that by

    themselves form a group.

    Examples of subgroups in P(3):

    E (E,A)(E,D,F)

    (E,B)

    (E,C)

    Theorem. If in a ?nite group, an element X is multiplied by itself enough

    times (n), the identity Xn = E is eventually recovered.

    Proof. If the group is ?nite, and any arbitrary element is multiplied by itself

    repeatedly, the product will eventually give rise to a repetition. For example,for P(3) which has six elements, seven multiplications must give a repetition.

    Let Y represent such a repetition:

    Y = Xp

    = Xq

    , where p>q. (1.5)

    Then let p = q + n so that

    Xp

    = Xq+n = Xq

    Xn = Xq

    = Xq

    E, (1.6)

    from which it follows that

    Xn = E. (1.7)

    

    De?nition 3. The order of an element ≡ the smallest value of n in the rela-

    tion Xn = E.

    We illustrate the order of an element using P(3) where:

    E is of order 1,? A,B,C are of order 2,? D,F are of order 3.1.5 Cosets 7

    De?nition 4. The period of an element X ≡ collection of elements E, X,X2, ...,Xn?1,where n is the order of the element. The period forms an

    Abelian subgroup.

    Some examples of periods based on the group P(3) are

    E,A

    E,B

    E,C

    E,D,F = E,D,D2 .

    (1.8)

    1.4 Rearrangement Theorem

    The rearrangement theorem is fundamental and basic to many theorems to

    be proven subsequently.

    Rearrangement Theorem. If E,A1,A2,...,Ah are the elements of

    agroup,andif Ak is an arbitrary group element, then the assembly of

    elements

    AkE,AkA1,...,AkAh (1.9)

    contains each element of the group once and only once.

    Proof. 1. We show ?rst that every element is contained.

    Let X be an arbitrary element. If the elements form a group there will

    be an element Ar = A?1

    k X.Then AkAr = AkA?1

    k X = X.Thuswecan

    always ?nd X after multiplication of the appropriate group elements.

    2. We now show that X occurs only once. Suppose that X appears twice

    in the assembly AkE,AkA1,...,AkAh,say X = AkAr = AkAs.Thenby

    multiplying on the left by A?1

    k we get Ar = As, which implies that two

    elements in the original group are identical, contrary to the original listing

    of the group elements.

    Because of the rearrangement theorem, every row and column of a multi-

    plication table contains each element once and only once. 

    1.5 Cosets

    In this section we will introduce the concept of cosets. The importance of

    cosets will be clear when introducing the factor group (Sect. 1.7). The cosets

    are the elements of a factor group, and the factor group is important for

    working with space groups (see Chap. 9).

    De?nition 5. If B is a subgroup of the group G,and X is an element of G,then the assembly EX,B1X,B2X,...,BgX is the right coset of B,where B

    consists of E,B1,B2,...,Bg.

    A coset need not be a subgroup. A coset wil l itself be a subgroup B if X is

    an element of B (by the rearrangement theorem).8 1 Basic Mathematical Background: Introduction

    Theorem. Two right cosets of given subgroup either contain exactly the same

    elements, or else have no elements in common.

    Proof. Clearly two right cosets either contain no elements in common or at

    least one element in common.We show that if there is one element in common,all elements are in common.

    Let BX and BY be two right cosets. If BkX = BY = one element that

    the two cosets have in common, then

    B?1

     Bk = YX?1

    (1.10)

    and YX?1 is in B, since the product on the left-hand side of (1.10) is in B.

    And also contained in B is EYX?1, B1YX?1, B2YX?1, ... , BgYX?1.Fur-

    thermore, according to the rearrangement theorem, these elements are, in

    fact, identical with B except for possible order of appearance. Therefore the

    elements of BY are identical to the elements of BYX?1X,whicharealso

    identical to the elements of BX so that all elements are in common. 

    We now give some examples of cosets using the group P(3). Let B = E,A be

    a subgroup. Then the right cosets of B are

    (E,A)E → E,A (E,A)C → C,F

    (E,A)A → A,E (E,A)D → D,B

    (E,A)B → B,D (E,A)F → F,C , (1.11)

    so that there are three distinct right cosets of (E,A), namely

    (E,A) which is a subgroup

    (B,D) which is not a subgroup

    (C,F) which is not a subgroup.

    Similarly there are three left cosets of (E,A) obtained by X(E,A):

    (E,A)

    (C,D)

    (B,F) .

    (1.12)

    To multiply two cosets, we multiply constituent elements of each coset in

    proper order. Such multiplication either yields a coset or joins two cosets. For

    example:

    (E,A)(B,D)=(EB,ED,AB,AD)=(B,D,D,B)=(B,D) . (1.13)

    Theorem. The order of a subgroup is a divisor of the order of the group.

    Proof. If an assembly of all the distinct cosets of a subgroup is formed (n of

    them), then n multiplied by the number of elements in a coset, C,isexactly1.6 Conjugation and Class 9

    the number of elements in the group. Each element must be included since

    cosets have no elements in common.

    For example, for the group P(3), the subgroup (E,A)isoforder2,the

    subgroup (E,D,F) is of order 3 and both 2 and 3 are divisors of 6, which is

    the order of P(3). 

    1.6 Conjugation and Class

    De?nition 6. An element B conjugate to A is by de?nition B ≡ XAX?1,where X is an arbitrary element of the group.

    For example,A = X?1

    BX = YBY ?1

    , where BX = XA and AY = YB.

    The elements of an Abelian group are all selfconjugate.

    Theorem. If B is conjugate to A and C is conjugate to B,then C is conjugate

    to A.

    Proof. By de?nition of conjugation, we can write

    B = XAX?1

    C = YBY?1

    .

    Thus, upon substitution we obtain

    C = YXAX?1

    Y ?1

    = YXA(YX)

    1

    .

    

    De?nition 7. A class is the totality of elements which can be obtained from

    a given group element by conjugation.

    For example in P(3), there are three classes:

    1. E;

    2. A,B,C;

    3. D,F.

    Consistent with this class designation is

    ABA?1

    = AF = C (1.14)

    DBD?1

    = DA = C. (1.15)

    Note that each class corresponds to a physically distinct kind of symmetry

    operation such as rotation of π about equivalent twofold axes, or rotation10 1 Basic Mathematical Background: Introduction

    of 2π3 about equivalent threefold axes. The identity symmetry element is

    always in a class by itself. An Abelian group has as many classes as elements.

    The identity element is the only class forming a group, since none of the other

    classes contain the identity.

    Theorem. All elements of the same class have the same order.

    Proof. The order of an element n is de?ned by An = E. An arbitrary conju-

    gate of A is B = XAX?1.Then Bn =(XAX?1)(XAX?1) ...n times gives

    XAnX?1 = XEX?1 = E.

    De?nition 8. Asubgroup B is self-conjugate (or invariant, or normal ) if

    XBX?1 is identical with B for all possible choices of X in the group.

    For example (E,D,F) forms a self-conjugate subgroup of P(3), but (E,A)

    does not. The subgroups of an Abelian group are self-conjugate subgroups.We

    will denote self-conjugate subgroups by N. To form a self-conjugate subgroup,it is necessary to include entire classes in this subgroup.

    De?nition 9. A group with no self-conjugate subgroups ≡ asimplegroup.

    Theorem. The right and left cosets of a self-conjugate subgroup N are the

    same.

    Proof. If Ni is an arbitrary element of the subgroup N, then the left coset is

    found by elements XNi = XNiX?1X = NjX,wheretherightcosetisformed

    by the elements NjX,where Nj = XNkX?1.

    For example in the group P(3), one of the right cosets is (E,D,F)A =

    (A,C,B) and one of the left cosets is A(E,D,F)=(A,B,C) and both cosets

    are identical except for the listing of the elements. 

    Theorem. The multiplication of the elements of two right cosets of a self-

    conjugate subgroup gives another right coset.

    Proof. Let NX and NY be two right cosets. Then multiplication of two right

    cosets gives

    (NX)(NY ) ? NiXNY = Ni(XN)Y

    = Ni(NmX)Y =(NiNm)(XY ) ?N(XY ) (1.16)

    and N(XY ) denotes a right coset. 

    The elements in one right coset of P(3) are (E,D,F)A =(A,C,B) while

    (E,D,F)D =(D,F,E) is another right coset. The product (A,C,B)(D,F,E)

    is (A,B,C) which is a right coset. Also the product of the two right cosets

    (A,B,C)(A,B,C)is(D,F,E) which is a right coset.1.8 Group Theory and Quantum Mechanics 11

    1.7 Factor Groups

    De?nition 10. The factor group (or quotient group) is constructed with re-

    spect to a self-conjugate subgroup as the collection of cosets of the self-

    conjugate subgroup, each coset being considered an element of the factor group.

    The factor group satis?es the four rules of Sect. 1.1 and is therefore a group:

    1. Multiplication – (NX)(NY )= NXY .

    2. Associative law – holds because it holds for the elements.

    3. Identity – EN,where E is the coset that contains the identity element.

    N is sometimes called a normal divisor.

    4. Inverse – (XN)(X?1N)=(NX)(X?1N)= N2 = EN.

    De?nition 11. The index of a subgroup ≡ total number of cosets = (order of

    group) (order of subgroup).

    The order of the factor group is the index of the self-conjugate subgroup.

    In Sect. 1.6 we saw that (E,D,F) forms a self-conjugate subgroup, N.

    The only other coset of this subgroup N is (A,B,C), so that the order of this

    factor group = 2. Let (A,B,C)= A and (E,D,F)= E be the two elements

    of the factor group. Then the multiplication table for this factor group is

    EA

    E EA

    A AE

    E is the identity element of this factor group. E and A are their own inverses.

    From this illustration you can see how the four group properties (see Sect. 1.1)

    apply to the factor group by taking an element in each coset, carrying out the

    multiplication of the elements and ?nding the coset of the resulting element.

    Note that this multiplication table is also the multiplication table for the

    group for the permutation of two objects P(2), i.e., this factor group maps

    one-on-one to the group P(2). This analogy between the factor group and

    P(2) gives insights into what the factor group is about.

    1.8 Group Theory and Quantum Mechanics

    We have now learned enough to start making connection of group theory to

    physical problems. In such problems we typically have a system described

    by a Hamiltonian which may be very complicated. Symmetry often allows us

    to make certain simpli?cations, without knowing the detailed Hamiltonian.

    To make a connection between group theory and quantum mechanics, we

    consider the group of symmetry operators ? PR which leave the Hamiltonian

    invariant. These operators ? PR are symmetry operations of the system and the

    PR operators commute with the Hamiltonian. The operators ? PR are said to12 1 Basic Mathematical Background: Introduction

    form the group of the Schr¨ odinger equation.If H and ? PR commute, and if ? PR

    is a Hermitian operator, then H and ? PR can be simultaneously diagonalized.

    We now show that these operators form a group. The identity element

    clearly exists (leaving the system unchanged). Each symmetry operator ? PR

    has an inverse ? P?1

    R to undo the operation ? PR and from physical considerations

    the element ? P?1

    R is also in the group. The product of two operators of the

    group is still an operator of the group, since we can consider these separately

    as acting on the Hamiltonian. The associative law clearly holds. Thus the

    requirements for forming a group are satis?ed.

    Whether the operators ? PR be rotations, re?ections, translations, or per-

    mutations, these symmetry operations do not alter the Hamiltonian or its

    eigenvalues. If Hψn = Enψn is a solution to Schr¨ odinger’s equation and H

    and ? PR commute, then

    PRHψn = ? PREnψn = H( ? PRψn)= En( ? PRψn) . (1.17)

    Thus ? PRψn is as good an eigenfunction of H as ψn itself. Furthermore, both

    ψn and ? PRψn correspond to the same eigenvalue En. Thus, starting with

    a particular eigenfunction, we can generate all other eigenfunctions of the same

    degenerate set (same energy) by applying all the symmetry operations that

    commute with the Hamiltonian (or leave it invariant). Similarly, if we consider

    the product of two symmetry operators, we again generate an eigenfunction

    of the Hamiltonian H

    PR ? PSH = H ? PR ? PS

    PR ? PSHψn = ? PR ? PSEnψn = En( ? PR ? PSψn)= H( ? PR ? PSψn) , (1.18)

    in which ? PR ? PSψn is also an eigenfunction of H. We also note that the action

    of ? PR on an arbitrary vector consisting of  eigenfunctions, yields a  × 

    matrix representation of ? PR that is in block diagonal form. The representation

    of physical systems, or equivalently their symmetry groups, in the form of

    matrices is the subject of the next chapter.

    Selected Problems

    1.1. (a) Show that the trace of an arbitrary square matrix X is invariant

    under a similarity (or equivalence) transformation UXU?1.

    (b) Given a set of matrices that represent the group G, denoted by D(R)(for

    all R in G), show that the matrices obtainable by a similarity transfor-

    mation UD(R)U?1 also are a representation of G.

    1.2. (a) Show that the operations of P(3) in (1.1) form a group, referring to

    the rules in Sect. 1.1.

    (b) Multiply the two left cosets of subgroup (E,A): (B,F)and(C,D), refer-

    ring to Sect. 1.5. Is the result another coset?1.8 Group Theory and Quantum Mechanics 13

    (c) Prove that in order to form a normal (self-conjugate) subgroup, it is nec-

    essary to include only entire classes in this subgroup.What is the physical

    consequence of this result?

    (d) Demonstrate that the normal subgroup of P(3) includes entire classes.

    1.3. (a) What are the symmetry operations for the molecule AB4,wherethe

    B atoms lie at the corners of a square and the A atom is at the center

    and is not coplanar with the B atoms.

    (b) Find the multiplication table.

    (c) List the subgroups. Which subgroups are self-conjugate?

    (d) List the classes.

    (e) Find the multiplication table for the factor group for the self-conjugate

    subgroup(s) of (c).

    1.4. The group de?ned by the permutations of four objects, P(4), is isomor-

    phic (has a one-to-one correspondence) with the group of symmetry opera-

    tions of a regular tetrahedron (Td). The symmetry operations of this group

    are su?ciently complex so that the power of group theoretical methods can be

    appreciated. For notational convenience, the elements of this group are listed

    below.

    e = (1234) g = (3124) m = (1423) s = (4213)

    a = (1243) h = (3142) n = (1432) t = (4231)

    b = (2134) i = (2314) o = (4123) u = (3412)

    c = (2143) j = (2341) p = (4132) v = (3421)

    d = (1324) k = (3214) q = (2413) w = (4312)

    f = (1342) l = (3241) r = (2431) y = (4321) .

    Here we have used a shorthand notation to denote the elements: for example

    j = (2341) denotes

    

    1234

    2341

    

    ,that is, the permutation which takes objects in the order 1234 and leaves them

    in the order 2341:

    (a) What is the product vw? wv?

    (b) List the subgroups of this group which correspond to the symmetry oper-

    ations on an equilateral triangle.

    (c) List the right and left cosets of the subgroup (e, a, k, l, s, t).

    (d) List all the symmetry classes for P(4), and relate them to symmetry op-

    erations on a regular tetrahedron.

    (e) Find the factor group and multiplication table formed from the self-

    conjugate subgroup (e, c, u, y). Is this factor group isomorphic to P(3)?2

    Representation Theory and Basic Theorems

    In this chapter we introduce the concept of a representation of an abstract

    group and prove a number of important theorems relating to irreducible rep-

    resentations, including the “Wonderful Orthogonality Theorem.” This math-

    ematical background is necessary for developing the group theoretical frame-

    work that is used for the applications of group theory to solid state physics.

    2.1 Important De?nitions

    De?nition 12. Two groups are isomorphic or homomorphic if there exists

    a correspondence between their elements such that

    A → ? A

    B → ? B

    AB → ? A ? B,where the plain letters denote elements in one group and the letters with carets

    denote elements in the other group. If the two groups have the same order

    (same number of elements), then they are isomorphic (one-to-one correspon-

    dence). Otherwise they are homomorphic (many-to-one correspondence).

    For example, the permutation group of three numbers P(3) is isomorphic

    to the symmetry group of the equilateral triangle and homomorphic to its

    factor group, as shown in Table 2.1. Thus, the homomorphic representations

    in Table 2.1 are unfaithful. Isomorphic representations are faithful, because

    they maintain the one-to-one correspondence.

    De?nition 13. A representation of an abstract group is a substitution group

    (matrix group with square matrices) such that the substitution group is homo-

    morphic (or isomorphic) to the abstract group. We assign a matrix D(A) to

    each element A of the abstract group such that D(AB)= D(A)D(B).16 2 Representation Theory and Basic Theorems

    Table 2.1. Table of homomorphic mapping of P(3) and its factor group

    permutation group element factor group

    E,D,F →E

    A,B,C →A

    The matrices of (1.4) are an isomorphic representation of the permutation

    group P(3). In considering the representation

    E

    D

    F

    → (1)

    A

    B

    C

    → (?1)

    the one-dimensional matrices (1) and (?1) are a homomorphic representa-

    tion of P(3) and an isomorphic representation of the factor group E,A (see

    Sect. 1.7). The homomorphic one-dimensional representation (1) is a repre-

    sentation for any group, though an unfaithful one.

    In quantum mechanics, the matrix representation of a group is important

    for several reasons. First of all, we will ?nd that an eigenfunction for a quan-

    tum mechanical operator will transform under a symmetry operation similar

    to the application of the matrix representing the symmetry operation on the

    matrix for the wave function. Secondly, quantum mechanical operators are

    usually written in terms of a matrix representation, and thus it is convenient

    to write symmetry operations using the same kind of matrix representa-

    tion. Finally, matrix algebra is often easier to manipulate than geometrical

    symmetry operations.

    2.2 Matrices

    De?nition 14. Hermitian matrices are de?ned by: ? A = A?, ? A? = A,or A? =

    A (where the symbol ? denotes complex conjugation, ～ denotes transposition,and ? denotes taking the adjoint )

    A =

    ?

    a11 a12 ···

    a21 a22 ···

    .

    .

    .

    .

    .

    .

    ? , (2.1)

    A =

    ?

    a11 a21 ···

    a12 a22 ···

    .

    .

    .

    .

    .

    .

    ? , (2.2)

    A?

    =

    ?

    a?

    11 a?

    21 ···

    a?

    12 a?

    22 ···

    .

    .

    .

    .

    .

    .

    ? . (2.3)2.3 Irreducible Representations 17

    Unitary matrices are de?ned by: ? A? = A? = A?1;

    Orthonormal matrices are de?ned by: ? A = A?1.

    De?nition 15. The dimensionality of a representation is equal to the dimen-

    sionality of each of its matrices, which is in turn equal to the number of rows

    or columns of the matrix.

    These representations are not unique. For example, by performing a similarity

    (or equivalence, or canonical) transformation UD(A)U?1 we generate a new

    set of matrices which provides an equally good representation. A simple phys-

    ical example for this transformation is the rotation of reference axes, such as

    (x, y, z)to(x

    ,y

    ,z). We can also generate another representation by taking

    one or more representations and combining them according to

    

    D(A) O

    O D

    (A)

    

    , (2.4)

    where O =(m×n) matrix of zeros, not necessarily a square zero matrix. The

    matrices D(A)and D

    (A) can be either two distinct representations or they

    can be identical representations.

    To overcome the di?culty of non-uniqueness of a representation with re-

    gard to a similarity transformation, we often just deal with the traces of the

    matrices which are invariant under similarity transformations, as discussed in

    Chap.3.The trace of a matrix is de?ned as the sum of the diagonal matrix

    elements. To overcome the di?culty of the ambiguity of representations in

    general, we introduce the concept of irreducible representations.

    2.3 Irreducible Representations

    Consider the representation made up of two distinct or identical representa-

    tions for every element in the group

    

    D(A) O

    O D

    (A)

    

    .

    This is a reducible representation because the matrix corresponding to each

    and every element of the group is in the same block form. We could now

    carry out a similarity transformation which would mix up all the elements so

    that the matrices are no longer in block form. But still the representation is

    reducible. Hence the de?nition:

    De?nition 16. If by one and the same equivalence transformation, all the

    matrices in the representation of a group can be made to acquire the same

    block form, then the representation is said to be reducible; otherwise it is

    irreducible. Thus, an irreducible representation cannot be expressed in terms

    of representations of lower dimensionality.18 2 Representation Theory and Basic Theorems

    We will now consider three irreducible representations for the permutation

    group P(3):

    EA B

    Γ1 : (1) (1) (1)

    Γ1 :(1) (?1) (?1)

    Γ2 :

    

    10

    01

    

    10

    01

      1

    2 ?

    √3

    2

    √3

    2 ?1

    2

    

    CD F

    Γ1 : (1) (1) (1)

    Γ1 :(?1) (1) (1)

    Γ2 :

     1

    2

    √3

    2 √3

    2 ?1

    2

    

    1

    2

    √3

    2

    √3

    2 ?1

    2

    

    1

    2 ?

    √3

    2 √3

    2 ?1

    2

    

    .

    (2.5)

    A reducible representation containing these three irreducible representations is

    EA B

    ΓR :

    ? ?

    1000

    0100

    0010

    0001

    ? ?

    ? ?

    10 00

    0 ?100

    00 ?10

    00 01

    ? ?

    ? ?

    10 0 0

    0 ?10 0

    00 1

    2 ?

    √3

    2

    00 ?

    √3

    2 ?1

    2

    ? ? ··· ,(2.6)

    where ΓR is of the form ?

    Γ1 0 O

    0 Γ1 O

    O O Γ2

    . (2.7)

    It is customary to list the irreducible representations contained in a reducible

    representation ΓR as

    ΓR = Γ1 + Γ1 + Γ2 . (2.8)

    In working out problems of physical interest, each irreducible representation

    describes the transformation properties of a set of eigenfunctions and corre-

    sponds to a distinct energy eigenvalue. Assume ΓR is a reducible represen-

    tation for some group G but an irreducible representation for some other

    group G

    .If ΓR contains the irreducible representations Γ1 + Γ1 + Γ2 as il-

    lustrated earlier for the group P(3), this indicates that some interaction is

    breaking up a fourfold degenerate level in group G

    into three energy levels in

    group G: two nondegenerate ones and a doubly degenerate one. Group theory

    does not tell us what these energies are, nor their ordering. Group theory

    only speci?es the symmetries and degeneracies of the energy levels. In gen-

    eral, the higher the symmetry, meaning the larger the number of symmetry

    operations in the group, the higher the degeneracy of the energy levels. Thus

    when a perturbation is applied to lower the symmetry, the degeneracy of the

    energy levels tends to be reduced. Group theory provides a systematic method

    for determining exactly how the degeneracy is lowered.2.4 The Unitarity of Representations 19

    Representation theory is useful for the treatment of physical problems be-

    cause of certain orthogonality theorems which we will now discuss. To prove

    the orthogonality theorems we need ?rst to prove some other theorems (in-

    cluding the unitarity of representations in Sect. 2.4 and the two Schur lemmas

    in Sects. 2.5 and 2.6).

    2.4 The Unitarity of Representations

    The following theorem shows that in most physical cases, the elements of

    a group can be represented by unitary matrices, which have the property of

    preserving length scales. This theorem is then used to prove lemmas leading

    to the proof of the “Wonderful Orthogonality Theorem,” which is a central

    theorem of this chapter.

    Theorem. Every representation with matrices having nonvanishing determi-

    nants can be brought into unitary form by an equivalence (similarity) trans-

    formation.

    Proof. By unitary form we mean that the matrix elements obey the relation

    (A?1)ij = A?

    ij = A?

    ji,where A is an arbitrary matrix of the representation.

    The proof is carried out by actually ?nding the corresponding unitary matrices

    if the Aij matrices are not already unitary matrices.

    Let A1,A2, ··· ,Ah denote matrices of the representation. We start by

    forming the matrix sum

    H =

    h 

    x=1

    AxA?

    x , (2.9)

    where the sum is over all the elements in the group and where the adjoint of

    a matrix is the transposed complex conjugate matrix (A?

    x)ij =(Ax)?

    ji.The

    matrix H is Hermitian because

    H?

    =

    

    x

    (AxA?

    x)

    =

    

    x

    AxA?

    x . (2.10)

    Any Hermitian matrix can be diagonalized by a suitable unitary transforma-

    tion. Let U be a unitary matrix made up of the orthonormal eigenvectors

    which diagonalize H to give the diagonal matrix d:

    d = U?1

    HU

    =

    

    x

    U?1

    AxA?

    xU

    =

    

    x

    U?1

    AxUU?1

    A?

    xU

    =

    

    x

    Ax ? A?

    x , (2.11)20 2 Representation Theory and Basic Theorems

    wherewede?ne ? Ax = U?1AxU for all x. The diagonal matrix d is a special

    kind of matrix and contains only real, positive diagonal elements since

    dkk =

    

    x

    

    j

    ( ? Ax)kj ( ? A?

    x)jk

    =

    

    x

    

    j

    ( ? Ax)kj ( ? Ax)

    kj

    =

    

    x

    

    j

    |( ? Ax)kj |

    2

    . (2.12)

    Out of the diagonal matrix d, one can form two matrices (d12 and d?12)

    such that

    d12

    ≡

    ?

    √d11 O √d22

    O .

    .

    .

    ? (2.13)

    and

    d?12

    ≡

    ?

    1 √d11

    O

    1 √d22

    O .

    .

    .

    ? , (2.14)

    where d12 and d?12 are real, diagonal matrices. We note that the generation

    of d?12 from d12 requires that none of the dkk vanish. These matrices clearly

    obey the relations

    (d12)

    = d12

    (2.15)

    (d?12)

    = d?12

    (2.16)

    (d12)(d12)= d (2.17)

    so that

    d12

    d?12

    = d?12

    d12

    = ? 1 = unit matrix . (2.18)

    From (2.11) we can also write

    d = d12

    d12

    =

    

    x

    Ax ? A?

    x . (2.19)

    We now de?ne a new set of matrices

    ? Ax ≡ d?12 ? Axd12

    (2.20)

    and

    A?

    x =(U?1

    AxU)

    = U?1

    A?

    xU (2.21)

    ? A?

    x =(d?12 ? Axd12)

    = d12 ?

    A?

    xd?12

    . (2.22)2.5 Schur’s Lemma (Part 1) 21

    We now show that the matrices

    ? Ax are unitary:

    ? Ax

    ? A?

    x =(d?12 ? Axd12)(d12 ? A?

    xd?12)

    = d?12 ? Axd ? A?

    xd?12

    = d?12

    

    y

    Ax ? Ay ? A?

    y

    A?

    xd?12

    = d?12

    

    y

    ( ? Ax ? Ay)( ? Ax ? Ay)

    d?12

    = d?12

    

    z

    Az

    A?

    z d?12

    (2.23)

    by the rearrangement theorem (Sect. 1.4). But from the relation

    d =

    

    z

    Az

    A?

    z (2.24)

    it follows that

    ? Ax

    ? A?

    x = ? 1, so that

    ? Ax is unitary.

    Therefore we have demonstrated how we can always construct a unitary

    representation by the transformation:

    ? Ax = d?12

    U?1

    AxUd12

    , (2.25)

    where

    H =

    h 

    x=1

    AxA?

    x (2.26)

    d =

    h 

    x=1

    Ax ? A?

    x , (2.27)

    and where U is the unitary matrix that diagonalizes the Hermitian matrix H

    and ? Ax = U?1AxU. 

    Note: On the other hand, not all symmetry operations can be represented by

    a unitary matrix; an example of an operation which cannot be represented by

    a unitary matrix is the time inversion operator (see Chap. 16). Time inversion

    symmetry is represented by an antiunitary matrix rather than a unitary ma-

    trix. It is thus not possible to represent all symmetry operations by a unitary

    matrix.

    2.5 Schur’s Lemma (Part 1)

    Schur’s lemmas (Parts 1 and 2) on irreducible representations are proved in

    order to prove the “Wonderful Orthogonality Theorem” in Sect. 2.7. We next

    prove Schur’s lemma Part 1.22 2 Representation Theory and Basic Theorems

    Lemma. A matrix which commutes with all matrices of an irreducible repre-

    sentation is a constant matrix, i.e., a constant times the unit matrix. There-

    fore, if a non-constant commuting matrix exists, the representation is re-

    ducible; if none exists, the representation is irreducible.

    Proof. Let M be a matrix which commutes with all the matrices of the rep-

    resentation A1,A2,...,Ah 

    MAx = AxM. (2.28)

    Take the adjoint of both sides of (2.28) to obtain

    A?

    xM?

    = M?

    A?

    x . (2.29)

    Since Ax can in all generality be taken to be unitary (see Sect. 2.4), multiply

    on the right and left of (2.29) by Ax to yield

    M?

    Ax = AxM?

    , (2.30)

    so that if M commutes with Ax so does M?, and so do the Hermitian matrices

    H1 and H2 de?ned by

    H1 = M +M?

    H2 = i(M ?M?) , (2.31)

    HjAx = AxHj , where j =1, 2 . (2.32)

    We will now show that a commuting Hermitian matrix is a constant matrix

    from which it follows that M = H1 ? iH2 is also a constant matrix.

    Since Hj (j =1, 2) is a Hermitian matrix, it can be diagonalized. Let U

    be the matrix that diagonalizes Hj (for example H1) to give the diagonal

    matrix d

    d = U?1

    HjU. (2.33)

    We now perform the unitary transformation on the matrices Ax of the rep-

    resentation ? Ax = U?1AxU. From the commutation relations (2.28), (2.29),and (2.32), a unitary transformation on all matrices HjAx = AxHj yields

    (U?1

    HjU)

      

    d

    (U?1

    AxU)

      

    Ax

    =(U?1

    AxU)

      

    Ax

    (U?1

    HjU)

      

    d

    . (2.34)

    So now we have a diagonal matrix d which commutes with all the matrices of

    the representation.We now show that this diagonal matrix d is a constant ma-

    trix, if all the ? Ax matrices (and thus also the Ax matrices) form an irreducible

    representation. Thus, starting with (2.34)

    d ? Ax = ? Axd (2.35)

    we take the ij element of both sides of (2.35)2.6 Schur’s Lemma (Part 2) 23

    dii( ? Ax)ij =( ? Ax)ij djj , (2.36)

    so that

    ( ? Ax)ij (dii ? djj ) = 0 (2.37)

    for all the matrices Ax.

    If dii 

    = djj , so that the matrix d is not a constant diagonal matrix, then

    ( ? Ax)ij must be 0 for all the ? Ax. This means that the similarity or unitary

    transformation U?1AxU has brought all the matrices of the representation

    Ax into the same block form, since any time dii 

    = djj all the matrices ( ? Ax)ij

    are null matrices. Thus by de?nition the representation Ax is reducible. But we

    have assumed the Ax to be an irreducible representation. Therefore ( ? Ax)ij 

    =0

    for all ? Ax,sothatitisnecessarythat dii = djj , and Schur’s lemma Part 1 is

    proved.

    2.6 Schur’s Lemma (Part 2)

    Lemma. If the matrix representations D(1)

    (A1),D(1)

    (A2),...,D(1)

    (Ah)

    and D(2)

    (A1),D(2)

    (A2),...,D(2)

    (Ah) are two irreducible representations

    of a given group of dimensionality 1 and 2, respectively, then, if there is

    amatrixof 1 columns and 2 rows M such that

    MD(1)

    (Ax)= D(2)

    (Ax)M (2.38)

    for all Ax,then M must be the null matrix (M = O)if 1 

    = 2.If 1 = 2,then either M = O or the representations D(1)

    (Ax) and D(2)

    (Ax) di?er from

    each other by an equivalence (or similarity) transformation.

    Proof. Since the matrices which form the representation can always be trans-

    formed into unitary form, we can in all generality assume that the matrices of ......