Unitary Symmetry and Combinatorics

The following sections are included:

• Preface

• Contents

• Notation

The following sections are included:

• Background and Viewpoint
  • Euclidean and Cartesian 3-space
  • Newtonian physics
  • Nonrelativistic quantum physics
  • Unitary frame rotations

• Abstract Angular Momentum
  • Brief background and history
  • One angular momentum: First and Second Fundamental Results
  • Two angular momenta: Third Fundamental Result

• SO(3, ℝ) and SU(2) Solid Harmonics
  • Inner products

• Combinatorial Features
  • Combinatorial definition of Wigner-Clebsch-Gordan coeffcients
  • Magic square realization of the addition of two angular momenta

• Kronecker Product of Solid Harmonics

• SU(n) Solid Harmonics
  • Definition and properties of SU(n) solid harmonics
  • MacMahon and Schwinger master theorems
  • Combinatorial proof of the multiplication property
  • Maclaurin monomials
  • Summary of relations

• Generalization to U(2)
  • Definition of U(2) solid harmonics
  • Basic multiplication properties
  • Indeterminate and derivative actions on the U(2) solid harmonics

The following sections are included:

• General Setting
  • Angular momentum state vectors of a composite system
  • Group actions in a composite system
  • Standard form of the Kronecker direct sum
  • Reduction of Kronecker products
  • Recoupling matrices

• Binary Coupling Theory
  • Binary trees
  • Standard labeling of binary trees
  • Generalized WCG coefficients defined in terms of binary trees
  • Binary coupled state vectors
  • Binary reduction of Kronecker products
  • Binary recoupling matrices
  • Triangle patterns and triangle coefficients
  • Racah coefficients
  • Recoupling matrices for n = 3
  • Recoupling matrices for n = 4
  • Structure of general triangle coefficients and recoupling matrices

• Classification of Recoupling Matrices

The following sections are included:

• Binary Trees and Trivalent Trees

• Nonisomorphic Trivalent Trees
  • Shape labels of binary trees

• Cubic Graphs Associated with Pairs of Trivalent Trees

• Cubic Graphs
  • Factoring properties of cubic graphs
  • Join properties of cubic graphs
  • Cubic graph matrices
  • Labeled cubic graphs
  • Summary and unsolved problems

The following sections are included:

• Pfaffians and Double Pfaffians

• The Skew-Symmetric Matrix of a Standard Labeled Binary Tree

• Triangle Monomials

• Generating Functions for Coupled Wave Functions

• Generating Functions for Recoupling Coefficients

• Special Cases of the Generating Function
  • Cubic graph geometry of the 6 – j and 9 – j coefficients

• Concluding Remarks on Angular Momentum Theory

The following sections are included:

• Overview

• Defining Relations for the D^λ–Polynomials
  • Another proof of the multiplication rule

• The D^λ–Polynomials Over l × n Variables

• Vector Space Aspects

• Fundamental Structural Relations and Principal Objectives
  • Structural relations
  • Principal objectives and tasks

The following sections are included:

• Introductory Remarks

• Action of Fundamental Shift Operators

• Arc Digraph Interpretation of the Fundamental Shift Operators

• Algebra of the Fundamental Shift Operators

• Abstract Hilbert Space and D^λ-Polynomials

• Shift Operator Polynomials

• Kronecker Product Reduction and Operator-valued Polynomials

• More on Explicit Operator Actions

The following sections are included:

• Combinatorial Formula for the C^λ(A) Matrices

• Reduction of the Kronecker Product D^p(Z) ⊗ D^λ(Z)

• Binary Tree Structure of the C^λ–Coefficients
  • The D^Λ–polynomials for partitions with two nonzero parts
  • Recurrence relation for the D^λ–polynomials

The following sections are included:

• Background and Review

• GL(n,ℂ) and its Unitary Subgroup U(n)

• Complete Set of Commuting Hermitian Operators
  • The Gelfand invariants

• Differential Operator Actions

• Eigenvalues of the Gelfand Invariants
  • A new class of symmetric functions
  • The general eigenvalues of the Gelfand invariants

The following sections are included:

• Introduction
  • A basis of irreducible tensor operators

• Unit Tensor Operators
  • Summary of properties of unit tensor operators
  • Explicit unit tensor operators

• Canonical Tensor Operators
  • Application of the factorization lemma to canonical tensor operators
  • Subgroup conditions and reduced matrix elements

• Properties of U(n) : U(n−1) Reduced Matrix Elements
  • Unit projective operators
  • The pattern calculus
  • Shift invariance of Kostka and Littlewood-Richardson numbers

• The Unitary Group U(3)
  • The U(3) canonical tensor operators

• Characteristic Null Spaces of U(3) Canonical Tensor Operators

• The U(3) : U(2) Unit Projective Operators
  • Coupling rules and Racah coefficients
  • Limit relations

The following sections are included:

• Groups
  • Group actions

• Rings
  • Rings of polynomials
  • Vector spaces of polynomials

• Abstract Hilbert Spaces
  • Inner product spaces
  • Linear operators
  • Inner products and linear operators
  • Orthonormalization methods
  • Matrix representations of linear operators

• Properties of Matrices
  • Properties of normal matrices
  • Inner product on the space of complex matrices
  • Exponentiated matrices
  • Lie bracket polynomials

• Tensor Product Spaces

• Vector Spaces of Polynomials with an Inner Product

• Group Representations
  • Irreducible representations of groups
  • Schur's lemmas

The following sections are included:

• Partitions and Tableaux
  • Restricted compositions
  • Linear ordering of sequences and partitions

• Young Frames and Young-Weyl Tableaux
  • Skew tableaux

• Gelfand-Tsetlin Patterns
  • Combinatorial origin of Gelfand-Tsetlin patterns
  • Group theoretical origin of Gelfand-Tsetlin patterns
  • Skew Gelfand-Tsetlin patterns
  • Notations
  • Triangular and skew Gelfand-Tsetlin patterns
  • Words and lattice permutations
  • Lattice permutations and Littlewood-Richardson numbers
  • Kostka and Littlewood-Richardson numbers

• Generating Functions and Relations
  • Notations
  • Counting relations
  • Generating relations of functions
  • Operator generated functions

• Multivariable Special Functions
  • Solid harmonics
  • Double harmonic functions in ℝ³
  • Hypergeometric functions
  • MacMahon's master theorem
  • Power of a determinant

• Symmetric Functions
  • Introduction
  • Four basic symmetric functions
  • Vandermonde determinants
  • Schur functions
  • Dual bases for symmetric functions

• Sylvester's Identity

• Algebraic Derivation of Weyl's Dimension Formula
  • Vandermonde determinants, Bernoulli polynomials, and Weyl's formula

• Other Topics
  • Alternating sign matrices
  • Binary trees, graphs, and digraphs
  • Umbral calculus and double tableau calculus
  • Special functions
  • Other generalizations

The following sections are included:

• Bibliography

• Index