Unitary symmetry and combinatorics

Contents......Page 14
Preface......Page 8
Notation......Page 22
1.1.1 Euclidean and Cartesian 3-space......Page 24
1.1.2 Newtonian physics......Page 32
1.1.3 Nonrelativistic quantumphysics......Page 33
1.1.4 Unitary frame rotations......Page 40
1.2.1 Brief background and history......Page 51
1.2.2 One angular momentum......Page 53
1.2.3 Two angular momenta......Page 58
1.3 SO(3,R) and SU(2) Solid Harmonics......Page 66
1.3.1 Inner products......Page 72
1.4.1 Combinatorial definition of Wigner-Clebsch-Gordan coefficients......Page 74
1.4.2 Magic square realization......Page 82
1.5 Kronecker Product of Solid Harmonics......Page 84
1.6.1 Definition and properties of SU(n) solid harmonics......Page 87
1.6.2 MacMahon and Schwinger master theorems......Page 89
1.6.3 Combinatorial proof of the multiplication property......Page 90
1.6.4 Maclaurin monomials......Page 94
1.6.5 Summary of relations......Page 97
1.7.1 Definition of U(2) solid harmonics......Page 98
1.7.2 Basic multiplication properties......Page 101
1.7.3 Indeterminate and derivative actions on the U(2) solid harmonics......Page 104
2.1.1 Angular momentum state vectors of a composite system......Page 106
2.1.2 Group actions in a composite system......Page 112
2.1.3 Standard form of the Kronecker direct sum......Page 113
2.1.4 Reduction of Kronecker products......Page 116
2.1.5 Recoupling matrices......Page 117
2.2 Binary Coupling Theory......Page 120
2.2.1 Binary trees......Page 123
2.2.2 Standard labeling of binary trees......Page 130
2.2.3 Generalized WCG coe.cients defined in terms of binary trees......Page 132
2.2.4 Binary coupled state vectors......Page 135
2.2.5 Binary reduction of Kronecker products......Page 137
2.2.6 Binary recouplingmatrices......Page 138
2.2.7 Triangle patterns and triangle coefficients......Page 142
2.2.8 Racah coefficients......Page 154
2.2.9 Recoupling matrices for n=3......Page 160
2.2.10 Recoupling matrices for n=4......Page 163
2.2.11 Structure of general triangle coe.cients and recoupling matrices......Page 175
2.3 Classification of Recoupling Matrices......Page 179
3.1 Binary Trees and Trivalent Trees......Page 186
3.2 Nonisomorphic Trivalent Trees......Page 195
3.2.1 Shape labels of binary trees......Page 207
3.3 Cubic Graphs and Trivalent Trees......Page 212
3.4 Cubic Graphs......Page 221
3.4.1 Factoring properties of cubic graphs......Page 223
3.4.2 Join properties of cubic graphs......Page 236
3.4.3 Cubic graph matrices......Page 244
3.4.4 Labeled cubic graphs......Page 249
3.4.5 Summary and unsolved problems......Page 251
4 Generating Functions......Page 252
4.1 Pfaffians and Double Pfaffians......Page 253
4.2 Skew-Symmetric Matrix......Page 255
4.3 Triangle Monomials......Page 261
4.4 Coupled Wave Functions......Page 262
4.5 Recoupling Coefficients......Page 264
4.6 Special Cases......Page 268
4.6.1 Cubic graph geometry of the 6 — j and 9 — j coefficients......Page 278
4.7 Concluding Remarks......Page 282
5.1 Overview......Page 284
5.2 Defining Relations......Page 291
5.2.1 Another proof of the multiplication rule......Page 300
5.3 Restriction to Fewer Variables......Page 302
5.4 Vector Space Aspects......Page 303
5.5.1 Structural relations......Page 306
5.5.2 Principal objectives and tasks......Page 307
6.1 Introductory Remarks......Page 308
6.2 Action of Fundamental Shift Operators......Page 309
6.3 Digraph Interpretation......Page 318
6.4 Algebra of Shift Operators......Page 324
6.5 Hilbert Space and Dλ.Polynomials......Page 327
6.6 Shift Operator Polynomials......Page 331
6.7 Kronecker Product Reduction......Page 335
6.8 More on Explicit Operator Actions......Page 341
7.1 The Cλ(A)Matrices......Page 348
7.2 Reduction of Dp(Z)  Dλ(Z)......Page 352
7.3 Binary Tree Structure: C.–Coefficients......Page 358
7.3.1 The Dλ.polynomials for partitions with two nonzero parts......Page 365
7.3.2 Recurrence relation for the Dλ.polynomials......Page 374
8.1 Background and Review......Page 378
8.2 GL(n,C) and its Unitary Subgroup U(n)......Page 381
8.3.1 The Gelfand invariants......Page 391
8.4 Differential Operator Actions......Page 393
8.5 Eigenvalues of the Gelfand Invariants......Page 394
8.5.1 A new class of symmetric functions......Page 396
8.5.2 The general eigenvalues of the Gelfand invariants......Page 398
9.1 Introduction......Page 402
9.1.1 A basis of irreducible tensor operators......Page 405
9.2 Unit Tensor Operators......Page 409
9.2.1 Summary of properties of unit tensor operators......Page 412
9.2.2 Explicit unit tensor operators......Page 415
9.3 Canonical Tensor Operators......Page 417
9.3.1 Application of the factorization lemma to canonical tensor operators......Page 422
9.3.2 Subgroup conditions and reduced matrix elements......Page 426
9.4 Properties of ReducedMatrix Elements......Page 431
9.4.1 Unit projective operators......Page 433
9.4.2 The pattern calculus......Page 438
9.4.3 Shift invariance of Kostka and Littlewood-Richardson numbers......Page 444
9.5.1 The U(3) canonical tensor operators......Page 445
9.6 The U(3) Characteristic Null Spaces......Page 448
9.7 The U(3) : U(2) Unit Projective Operators......Page 455
9.7.1 Coupling rules and Racah coefficients......Page 456
9.7.2 Limit relations......Page 463
10.1 Groups......Page 470
10.1.1 Group actions......Page 472
10.2 Rings......Page 474
10.2.1 Rings of polynomials......Page 475
10.2.2 Vector spaces of polynomials......Page 478
10.3.1 Inner product spaces......Page 479
10.3.2 Linear operators......Page 481
10.3.3 Inner products and linear operators......Page 483
10.3.4 Orthonormalization methods......Page 484
10.3.5 Matrix representations of linear operators......Page 487
10.4 Properties ofMatrices......Page 488
10.4.1 Properties of normalmatrices......Page 490
10.4.2 Inner product on the space of complex matrices......Page 491
10.4.3 Exponentiated matrices......Page 492
10.4.4 Lie bracket polynomials......Page 499
10.5 Tensor Product Spaces......Page 509
10.6 Vector Spaces of Polynomials......Page 511
10.7 Group Representations......Page 517
10.7.1 Irreducible representations of groups......Page 522
10.7.2 Schur’s lemmas......Page 524
11 Compendium B: Combinatorial Objects......Page 528
11.1 Partitions and Tableaux......Page 529
11.1.1 Restricted compositions......Page 535
11.1.2 Linear ordering of sequences and partitions......Page 536
11.2 Young Frames and Tableaux......Page 537
11.2.1 Skew tableaux......Page 543
11.3.1 Combinatorial origin of Gelfand-Tsetlin patterns......Page 546
11.3.3 Skew Gelfand-Tsetlin patterns......Page 551
11.3.4 Notations......Page 555
11.3.5 Triangular and skew Gelfand-Tsetlin patterns......Page 556
11.3.6 Words and lattice permutations......Page 558
11.3.7 Lattice permutations and Littlewood-Richardson numbers......Page 561
11.3.8 Kostka and Littlewood-Richardson numbers......Page 573
11.4 Generating Functions and Relations......Page 579
11.4.1 Notations......Page 580
11.4.2 Counting relations......Page 581
11.4.3 Generating relations of functions......Page 582
11.4.4 Operator generated functions......Page 584
11.5 Multivariable Special Functions......Page 586
11.5.1 Solid harmonics......Page 587
11.5.3 Hypergeometric functions......Page 588
11.5.5 Power of a determinant......Page 591
11.6.1 Introduction......Page 593
11.6.2 Four basic symmetric functions......Page 594
11.6.3 Vandermonde determinants......Page 595
11.6.4 Schur functions......Page 596
11.6.5 Dual bases for symmetric functions......Page 601
11.7 Sylvester’s Identity......Page 603
11.8.1 Vandermonde determinants,Ber noulli polynomials, andWeyl’s formula......Page 605
11.9.1 Alternating sign matrices......Page 608
11.9.3 Umbral calculus and double tableau calculus......Page 618
11.9.5 Other generalizations......Page 619
Bibliography......Page 620
Index......Page 634