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CHAPTER I GROUPS AND MATRICES PAGE 1. The group concept . . .. . . 1 2. The symmetric group . . . . . 5 3. The full linear group . . . . . 11 4. Group representations . . . . . . . 16 5. Representation of the symmetric group by permutation matrices 17 6. Linear metric spaces . . . . . . . 17 7. Canonical forms for matrices . . . . 24 8. The rational one-dimensional representations of the full linear group . . . .30 Appendix to Chapter I: The canonical form for a matrix under transformation by an element of the full linear group . 34 CHAPTER II REDUCIBILITY 1. Reducibility . . . . . . . . 38 2. The analysis of any rational integral representation of the full linear group into a sum of homogeneous representations . 43 3. Schur's Lemma . . . 47 4. Burnside's Theorem . . . . . . . 48 5. Bounded representations of a given group . . . . 52 Appendix to Chapter II: The volume and mean center of a convex point set . . . . . . . . 64 CHAPTER III GROUP CHARACTERS 1. The Kronecker product . . . . . . . 68 2. The orthogonality relations for a finite group . . . 78 3. The characters of any representation of a finite group . . 81 CONTENTS PAGE 4. The number of non-equivalent irreducible representations of any finite group . . . . . 83 5. Symmetric linear operators . . . . . 85 CHAPTER IV THE SYMMETRIC GROUP 1. Cosets of a subgroup . . . . . . . 91 2. The characteristics of a finite group . . . 95 3. The direct product of two or more groups . . . 100 4. The principal characteristic of the symmetric group . . 103 5. The simple characteristics of the symmetric group . . 109 6. Associated irreducible representations of the symmetric group 120 7. The homogeneous rational integral representations of the full linear group . . . . . . 123 CHAPTER V THE CHARACTERS OF THE SYMMETRIC GROUP 1. The construction of the character tables for the symmetric groups . . . . 132 2. Formulae giving those characters of the symmetric group on m letters which are attached to two and three element parti- tions of m in terms of the class numbers (a) . . 143 3. The analysis of the reducible representations A(X) of the sym- metric group on m letters . .. . 150 4. The analysis of the direct product of irreducible representa- tions of the symmetric groups . 155 CHAPTER VI THE ALTERNATING GROUP 1. The classes of the alternating group . . . 168 2. The simple characteristics of the alternating group . . 171 CHAPTER VII LINEAR GROUPS PAGThe rational representations of the unimodular group 177 2. The rational representations of the extended unimodular group 180 3. The rationaluous representations of the extended unimodular group 180 3. The continuous representations of the unimodul linear group an183 4. The continuous representations of the real linear group and of the full linear group . . . . . . 196 CHAPTER VIII GROUP INTEGRATION 1. The volume element of the real linear group . . 202 2. The volume element of subgroups of the real linear group . 209 3. The orthogonality relations . . . . . 215 4. The element of volume of the full linear group and of its sub- groups . .. .217 5. The characteristic matrices of a subgroup of the full linear group . . . .219 CHAPTER IX THE ORTHOGONAL GROUP 1. The canonical representative of a class of the real orthogonal group . . . . 226 2. The element of volume of the n-dimensional rotation group 230 3. Integration over the n-dimensional real orthogonal group .237 4. The irreducible continuous representations of the real ortho- gonal group . . . . . . 242 5. The irreducible representations of the n-dimensional rotation group .. ..260 6. The analysis of the Kronecker product of irreducible repre- sentations of the real orthogonal group . . . 269 7. The modification rules for the n-dimensional orthogonal group 282 CONTENTS PAGE 8. The analysis of the representations of the real orthogonal group which are furnished, by the principle of selection, by the irreducible representations of the full linear group . . 285 9. The analysis of the representation of R'-_1 which is induced by the irreducible representation P,) of R' . . . 286 10. The characters of those irreducible representations of R2k which appear in the analysis of the representation of R2k which is induced by the irreducible representation Irx), Ax > 0, of R'2k 288 11. Alternative form for the simple characters of the real ortho- gonal group of odd dimension . . . . 294 CHAPTER X SPIN REPRESENTATIONS OF THE ROTATION GROUP 1. The two-valued representations of the three-dimensional rota- tion group . . . 296 2. Two-valued representations of the n-dimensional rotation group 299 3. The character of the spin representation of the n-dimensional real orthogonal group . . . . . . 306 4. The two-valued representations of the n-dimensional rotation group . . . . . . . . 311 5. The topology of the real n-dimensional rotation group .318 6. The covering group of the real n-dimensional rotation group 325 CHAPTER XI THE CRYSTALLOGRAPHIC GROUPS 1. The finite subgroups of the three-dimensional rotation group 328 2. The crystallographic groups . . . . . 336 3. The character tables of the 32 crystallographic groups .346 4. The symmetrized Kronecker square of an irreducible repre- sentation of a crystallographic group . . . . 350 CHAPTER XII THE LORENTZ GROUP PAGE 1. The four-dimensional Lorentz group . . . . . 352 2. The two-valued two-dimensional unimodular representation of G . 353 3. The theory of semi-vectors 358 4. The derivation of irreducible representations of the Lorentz group from irreducible representations of the attached orthogonal group .360 REFERENCE. . . . . .363 INDEX . . . . . . . . . .367