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Unitary symmetry and combinatorics

Louck, James D.

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Unitary symmetry and combinatorics

Record Type:	Electronic resources : Monograph/item
Title/Author:	Unitary symmetry and combinatorics/ James D. Louck.
Author:	Louck, James D.
Published:	Singapore ;World Scientific Pub. Co., : c2008.,
Description:	1 online resource (xxi, 619 p.) :ill.
[NT 15003449]:	1. Quantum angular momentum. 1.1. Background and viewpoint. 1.2 Abstract angular momentum. 1.3. SO(3,[symbol]) and SU(2) solid harmonics. 1.4. Combinatorial features. 1.5. Kronecker product of solid harmonics. 1.6. SU(n) solid harmonics. 1.7. Generalization to U(2) -- 2. Composite systems. 2.1. General setting. 2.2. Binary coupling theory. 2.3. Classification of recoupling matrices -- 3. Graphs and adjacency diagrams. 3.1. Binary trees and trivalent trees. 3.2. Nonisomorphic trivalent trees. 3.3. Cubic graphs and trivalent trees. 3.4. Cubic graphs -- 4. Generating functions. 4.1. Pfaffians and double Pfaffians. 4.2. Skew-symmetric matrix. 4.3. Triangle monomials. 4.4. Coupled wave functions. 4.5. Recoupling coefficients. 4.6. Special cases. 4.7. Concluding remarks -- 5. The [symbol]-polynomials: form. 5.1. Overview. 5.2. Defining relations. 5.3. Restriction to fewer variables. 5.4. Vector space aspects. 5.5. Fundamental structural relations
[NT 15003449]:	6. Operator actions in Hilbert space. 6.1. Introductory remarks. 6.2. Action of fundamental shift operators. 6.3. Digraph interpretation. 6.4. Algebra of shift operators. 6.5. Hilbert space and [symbol]-polynomials. 6.6. Shift operator polynomials. 6.7. Kronecker product reduction. 6.8. More on explicit operator actions -- 7. The [symbol]-polynomials: structure. 7.1. The [symbol] matrices. 7.2. Reduction of [symbol]. 7.3. Binary tree structure: [symbol]-coefficients -- 8. The general linear and unitary groups. 8.1. Background and review. 8.2. GL(n,[symbol]) and its unitary subgroup U(n). 8.3. Commuting Hermitian observables. 8.4. Differential operator actions. 8.5. Eigenvalues of the Gelfand invariants -- 9. Tensor operator theory. 9.1. Introduction. 9.2. Unit tensor operators. 9.3. Canonical tensor operators. 9.4. Properties of reduced matrix elements. 9.5. The unitary group U(3). 9.6. The U(3) characteristic null spaces. 9.7. The U(3) : U(2) unit projective operators
[NT 15003449]:	10. Compendium A. Basic algebraic objects. 10.1. Groups. 10.2. Rings. 10.3. Abstract Hilbert spaces. 10.4. Properties of matrices. 10.5. Tensor product spaces. 10.6. Vector spaces of polynomials. 10.7. Group representations -- 11. Compendium B: combinatorial objects. 11.1. Partitions and tableaux. 11.2. Young frames and tableaux. 11.3. Gelfand-Tsetlin patterns. 11.4. Generating functions and relations. 11.5. Multivariable special functions. 11.6. Symmetric functions. 11.7. Sylvester's identity. 11.8. Derivation of Weyl's dimension formula. 11.9. Other topics.
Subject:	Combinatorial analysis. -
Online resource:	http://www.worldscientific.com/worldscibooks/10.1142/6863#t=toc
ISBN:	9789812814739 (electronic bk.)

Unitary symmetry and combinatorics
Louck, James D.

Unitary symmetry and combinatorics[electronic resource] /James D. Louck. - Singapore ;World Scientific Pub. Co.,c2008. - 1 online resource (xxi, 619 p.) :ill.

Includes bibliographical references (p. 597-609) and index.

1. Quantum angular momentum. 1.1. Background and viewpoint. 1.2 Abstract angular momentum. 1.3. SO(3,[symbol]) and SU(2) solid harmonics. 1.4. Combinatorial features. 1.5. Kronecker product of solid harmonics. 1.6. SU(n) solid harmonics. 1.7. Generalization to U(2) -- 2. Composite systems. 2.1. General setting. 2.2. Binary coupling theory. 2.3. Classification of recoupling matrices -- 3. Graphs and adjacency diagrams. 3.1. Binary trees and trivalent trees. 3.2. Nonisomorphic trivalent trees. 3.3. Cubic graphs and trivalent trees. 3.4. Cubic graphs -- 4. Generating functions. 4.1. Pfaffians and double Pfaffians. 4.2. Skew-symmetric matrix. 4.3. Triangle monomials. 4.4. Coupled wave functions. 4.5. Recoupling coefficients. 4.6. Special cases. 4.7. Concluding remarks -- 5. The [symbol]-polynomials: form. 5.1. Overview. 5.2. Defining relations. 5.3. Restriction to fewer variables. 5.4. Vector space aspects. 5.5. Fundamental structural relations

This monograph integrates unitary symmetry and combinatorics, showing in great detail how the coupling of angular momenta in quantum mechanics is related to binary trees, trivalent trees, cubic graphs, MacMahon's master theorem, and other basic combinatorial concepts. A comprehensive theory of recoupling matrices for quantum angular momentum is developed. For the general unitary group, polynomial forms in many variables called matrix Schur functions have the remarkable property of giving all irreducible representations of the general unitary group and are the basic objects of study. The structure of these irreducible polynomials and the reduction of their Kronecker product is developed in detail, as is the theory of tensor operators.

ISBN: 9789812814739 (electronic bk.)Subjects--Topical Terms:

523878
Combinatorial analysis.

LC Class. No.: QA167

Dewey Class. No.: 511.6

Unitary symmetry and combinatorics
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245 1 0 $a Unitary symmetry and combinatorics $h [electronic resource] / $c James D. Louck.
260 $a Singapore ; $a Hackensack, N.J. : $b World Scientific Pub. Co., $c c2008.
300 $a 1 online resource (xxi, 619 p.) : $b ill.
504 $a Includes bibliographical references (p. 597-609) and index.
505 0 $a 1. Quantum angular momentum. 1.1. Background and viewpoint. 1.2 Abstract angular momentum. 1.3. SO(3,[symbol]) and SU(2) solid harmonics. 1.4. Combinatorial features. 1.5. Kronecker product of solid harmonics. 1.6. SU(n) solid harmonics. 1.7. Generalization to U(2) -- 2. Composite systems. 2.1. General setting. 2.2. Binary coupling theory. 2.3. Classification of recoupling matrices -- 3. Graphs and adjacency diagrams. 3.1. Binary trees and trivalent trees. 3.2. Nonisomorphic trivalent trees. 3.3. Cubic graphs and trivalent trees. 3.4. Cubic graphs -- 4. Generating functions. 4.1. Pfaffians and double Pfaffians. 4.2. Skew-symmetric matrix. 4.3. Triangle monomials. 4.4. Coupled wave functions. 4.5. Recoupling coefficients. 4.6. Special cases. 4.7. Concluding remarks -- 5. The [symbol]-polynomials: form. 5.1. Overview. 5.2. Defining relations. 5.3. Restriction to fewer variables. 5.4. Vector space aspects. 5.5. Fundamental structural relations
505 0 $a 6. Operator actions in Hilbert space. 6.1. Introductory remarks. 6.2. Action of fundamental shift operators. 6.3. Digraph interpretation. 6.4. Algebra of shift operators. 6.5. Hilbert space and [symbol]-polynomials. 6.6. Shift operator polynomials. 6.7. Kronecker product reduction. 6.8. More on explicit operator actions -- 7. The [symbol]-polynomials: structure. 7.1. The [symbol] matrices. 7.2. Reduction of [symbol]. 7.3. Binary tree structure: [symbol]-coefficients -- 8. The general linear and unitary groups. 8.1. Background and review. 8.2. GL(n,[symbol]) and its unitary subgroup U(n). 8.3. Commuting Hermitian observables. 8.4. Differential operator actions. 8.5. Eigenvalues of the Gelfand invariants -- 9. Tensor operator theory. 9.1. Introduction. 9.2. Unit tensor operators. 9.3. Canonical tensor operators. 9.4. Properties of reduced matrix elements. 9.5. The unitary group U(3). 9.6. The U(3) characteristic null spaces. 9.7. The U(3) : U(2) unit projective operators
505 0 $a 10. Compendium A. Basic algebraic objects. 10.1. Groups. 10.2. Rings. 10.3. Abstract Hilbert spaces. 10.4. Properties of matrices. 10.5. Tensor product spaces. 10.6. Vector spaces of polynomials. 10.7. Group representations -- 11. Compendium B: combinatorial objects. 11.1. Partitions and tableaux. 11.2. Young frames and tableaux. 11.3. Gelfand-Tsetlin patterns. 11.4. Generating functions and relations. 11.5. Multivariable special functions. 11.6. Symmetric functions. 11.7. Sylvester's identity. 11.8. Derivation of Weyl's dimension formula. 11.9. Other topics.
520 $a This monograph integrates unitary symmetry and combinatorics, showing in great detail how the coupling of angular momenta in quantum mechanics is related to binary trees, trivalent trees, cubic graphs, MacMahon's master theorem, and other basic combinatorial concepts. A comprehensive theory of recoupling matrices for quantum angular momentum is developed. For the general unitary group, polynomial forms in many variables called matrix Schur functions have the remarkable property of giving all irreducible representations of the general unitary group and are the basic objects of study. The structure of these irreducible polynomials and the reduction of their Kronecker product is developed in detail, as is the theory of tensor operators.
650 0 $a Combinatorial analysis. $3 523878
650 0 $a Eightfold way (Nuclear physics) $3 940522
856 4 0 $u http://www.worldscientific.com/worldscibooks/10.1142/6863#t=toc