《The Theory of Group Characters and Matrix Representations of Groups Second Edition》

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Ⅰ．MATRICES1

1．1．Linear transformations1

1．2．Matrices2

1．3．The transform of a matrix4

1．4．Rectangular matrices and vectors5

1．5．The characteristic equation of a matrix6

1．6．The classical canonical form of a matrix7

1．7．The classical canonical form;multiple characteristic roots8

1．8．Various properties of matrices14

1．9．Unitary and orthogonal matrices15

Ⅱ．ALGEBRAS22

2．1．Definition of an algebra over the complex numbers22

2．2．Change of basis and the regular matrix representation23

2．3．Simple matrix algebras25

2．4．Examples of associative algebras25

2．5．Linear sets and sub-algebras26

2．6．Modulus,idempotent and nilpotent elements26

2．7．The reduced characteristic equation27

2．8．Reduction of an algebra relative to an idempotent29

2．9．The trace of an element31

Ⅲ．GROUPS32

3．1．Definition of a group32

3．2．Subgroups33

3．3．Examples of groups34

3．4．Permutation groups36

3．5．The alternating group37

3．6．Classes of conjugate elements38

3．7．Conjugate and self-conjugate subgroups40

3．8．The representations of an abstract group as a permutation group41

Ⅳ．THE FROBENIUS ALGEBRA43

4．1．Groups and algebras43

4．2．The group characters45

4．3．Matrix representations and group matrices48

4．4．Characteristic units56

4．5．The relations between the characters of a group and those of a subgroup57

Ⅴ．THE SYMMETRIC GROUP59

5．1．Partitions59

5．2．Frobenius＇s formula for the characters of the symmetric group61

5．3．Characters and lattices67

5．4．Primitive characteristic units and Young tableaux71

Ⅵ．IMMANANTS AND S-FUNCTIONS81

6．1．Immanants of a matrix81

6．2．Schur functions82

6．3．Properties of S-functions87

6．4．Generating functions and further properties of S-functions98

6．5．Relations between immanants and S-functions118

Ⅶ．S-FUNCTIONS OF SPECIAL SERIES122

7．1．The function φ(q,x)122

7．2．The functions (1－x)-n and (1－xr)-m126

7．3．S-functions associated with f(xr)131

Ⅷ．THE CALCULATION OF THE CHARACTERS OF THE SYMMETRIC GROUP137

8．1．Frobenius＇s formula137

S-functions of special series138

Recurrence relations140

Congruences142

Classes for which the orders of the cycles have a common factor143

Graphs and lattices146

Orthogonal properties146

Ⅸ．GROUP CHARACTERS AND THE STRUCTURE OF GROUPS147

9．1．The compound character associated with a subgroup147

9．2．Deduction of the characters of a subgroup from those of the group150

9．3．Determination of subgroups:necessary criteria that a compound character should correspond to a permutation representation of the group155

9．4．The properties of groups and character tables159

9．5．Transitivity164

9．6．Invariant subgroups171

Ⅹ．CONTINUOUS MATRIX GROUPS AND INVARIANT MATRICES178

10．1．Invariant matrices178

10．2．The classical canonical form of an invariant matrix193

10．3．Application to invariant theory203

Ⅺ．GROUPS OF UNITARY MATRICES210

11．1．Introductory210

11．2．Fundamental formula for integration over the group manifold211

11．3．Simplification of integration formulae for class functions217

11．4．Verification of the orthogonal properties of the characters of the unitary group222

11．5．Orthogonal matrices and the rotation groups223

11．6．Relations between the characters of D and D＇225

11．7．Integration formulae connected with D and D＇227

11．8．The characters of the orthogonal group233

11．9．Alternative forms for the characters of the orthogonal group238

11．10．The difference characters of the rotation group245

11．11．The spin representations of the orthogonal group248

11．12．Complex orthogonal matrices and groups of matrices with a quadratic invariant260

APPENDIX265

Tables of Characters of the Symmetric Groups265

Tables of Characters of Transitive Subgroups．Alternating Groups272

General Cyclic Group of Order n273

Other Transitive Subgroups273

Some Recent Developments285

BIBLIOGRAPHY301

SUPPLEMENTARY BIBLIOGRAPHY306

INDEX309