Applications of Unitary Symmetry and Combinatorics

Book excerpt: 1. Composite quantum systems. 1.1. Introduction. 1.2. Angular momentum state vectors of a composite system. 1.3. Standard form of the Kronecker direct sum. 1.4. Recoupling matrices. 1.5. Preliminary results on doubly stochastic matrices and permutation matrices. 1.6. Relationship between doubly stochastic matrices and density matrices in angular momentum theory -- 2. Algebra of permutation matrices. 2.1. Introduction. 2.2. Basis sets of permutation matrices -- 3. Coordinates of A in basis [symbol]. 3.1. Notations. 3.2. The A-expansion rule in the basis [symbol]. 3.3. Dual matrices in the basis set [symbol](e, p). 3.4. The general dual matrices in the basis [symbol](e, p) -- 4. Further applications of permutation matrices. 4.1. Introduction. 4.2. An algebra of young operators. 4.3. Matrix Schur functions. 4.4. Real orthogonal irreducible representations of S[symbol]. 4.5. Left and right regular representations of finite groups -- 5. Doubly stochastic matrices in angular momentum theory. 5.1. Introduction. 5.2. Abstractions and interpretations. 5.3. Permutation matrices as doubly stochastic. 5.4 The doubly stochastic matrix for a single system with angular momentum J. 5.5. Doubly stochastic matrices for composite angular momentum systems. 5.6. Binary coupling of angular momenta. 5.7. State vectors : Uncoupled and coupled. 5.8. General binary tree couplings and doubly stochastic matrices -- 6. Magic squares. 6.1. Review. 6.2. Magic squares and addition of angular momenta. 6.3. Rational generating function of H[symbol](r) -- 7. Alternating sign matrices. 7.1. Introduction. 7.2. Standard Gelfand-Tsetlin patterns. 7.3. Strict Gelfand-Tsetlin patterns for [symbol] = (nn-1 ... 21). 7.4. Sign-reversal-shift invariant polynomials. 7.5. The requirement of zeros. 7.6. The incidence matrix formulation -- 8. The Heisenberg magnetic ring. 8.1. Introduction. 8.2. Matrix elements of H in the uncoupled and coupled bases. 8.3. Exact solution of the Heisenberg ring magnet for n = 2,3,4. 8.4. The Heisenberg Ring Hamiltonian : Even n. 8.5. The Heisenberg Ring Hamiltonian : Odd n. 8.6. Recount, synthesis, and critique. 8.7 Action of the cyclic group. 8.8. Concluding remarks