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Ⅰ．MATRICES 1

1．1．Linear transformations 1

1．2．Matrices 2

1．3．The transform of a matrix 4

1．4．Rectangular matrices and vectors 5

1．5．The characteristic equation of a matrix 6

1．6．The classical canonical form of a matrix 7

1．7．The classical canonical form;multiple characteristic roots 8

1．8．Various properties of matrices 14

1．9．Unitary and orthogonal matrices 15

Ⅱ．ALGEBRAS 22

2．1．Definition of an algebra over the complex numbers 22

2．2．Change of basis and the regular matrix representation 23

2．3．Simple matrix algebras 25

2．4．Examples of associative algebras 25

2．5．Linear sets and sub-algebras 26

2．6．Modulus,idempotent and nilpotent elements 26

2．7．The reduced characteristic equation 27

2．8．Reduction of an algebra relative to an idempotent 29

2．9．The trace of an element 31

Ⅲ．GROUPS 32

3．1．Definition of a group 32

3．2．Subgroups 33

3．3．Examples of groups 34

3．4．Permutation groups 36

3．5．The alternating group 37

3．6．Classes of conjugate elements 38

3．7．Conjugate and self-conjugate subgroups 40

3．8．The representations of an abstract group as a permutation group 41

Ⅳ．THE FROBENIUS ALGEBRA 43

4．1．Groups and algebras 43

4．2．The group characters 45

4．3．Matrix representations and group matrices 48

4．4．Characteristic units 56

4．5．The relations between the characters of a group and those of a subgroup 57

Ⅴ．THE SYMMETRIC GROUP 59

5．1．Partitions 59

5．2．Frobenius＇s formula for the characters of the symmetric group 61

5．3．Characters and lattices 67

5．4．Primitive characteristic units and Young tableaux 71

Ⅵ．IMMANANTS AND S-FUNCTIONS 81

6．1．Immanants of a matrix 81

6．2．Schur functions 82

6．3．Properties of S-functions 87

6．4．Generating functions and further properties of S-functions 98

6．5．Relations between immanants and S-functions 118

Ⅶ．S-FUNCTIONS OF SPECIAL SERIES 122

7．1．The function φ(q,x) 122

7．2．The functions (1－x)-n and (1－xr)-m 126

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

Ⅷ．THE CALCULATION OF THE CHARACTERS OF THE SYMMETRIC GROUP 137

8．1．Frobenius＇s formula 137

S-functions of special series 138

Recurrence relations 140

Congruences 142

Classes for which the orders of the cycles have a common factor 143

Graphs and lattices 146

Orthogonal properties 146

Ⅸ．GROUP CHARACTERS AND THE STRUCTURE OF GROUPS 147

9．1．The compound character associated with a subgroup 147

9．2．Deduction of the characters of a subgroup from those of the group 150

9．3．Determination of subgroups:necessary criteria that a compound character should correspond to a permutation representation of the group 155

9．4．The properties of groups and character tables 159

9．5．Transitivity 164

9．6．Invariant subgroups 171

Ⅹ．CONTINUOUS MATRIX GROUPS AND INVARIANT MATRICES 178

10．1．Invariant matrices 178

10．2．The classical canonical form of an invariant matrix 193

10．3．Application to invariant theory 203

Ⅺ．GROUPS OF UNITARY MATRICES 210

11．1．Introductory 210

11．2．Fundamental formula for integration over the group manifold 211

11．3．Simplification of integration formulae for class functions 217

11．4．Verification of the orthogonal properties of the characters of the unitary group 222

11．5．Orthogonal matrices and the rotation groups 223

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 group 233

11．9．Alternative forms for the characters of the orthogonal group 238

11．10．The difference characters of the rotation group 245

11．11．The spin representations of the orthogonal group 248

11．12．Complex orthogonal matrices and groups of matrices with a quadratic invariant 260

APPENDIX 265

Tables of Characters of the Symmetric Groups 265

Tables of Characters of Transitive Subgroups．Alternating Groups 272

General Cyclic Group of Order n 273

Other Transitive Subgroups 273

Some Recent Developments 285

BIBLIOGRAPHY 301

SUPPLEMENTARY BIBLIOGRAPHY 306

INDEX 309