Richard J. Mathar
The Clebsch--Gordan coefficients of the Kronecker products of the irreducible representations of the Quaternion Group Q8, of the Generalized Quaternion Groups Q16 and Q32, and of the factor group (Q32 X Q32)/{(1,1),(-1,-1)} are computed as eigenvectors of a well-known matrix of triple-products of the irreducible representations.
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We provide an algorithm of computing Clebsch-Gordan coefficients for irreducible representations, with integer weights, of the rotation group SO(3) and demonstrate the convenience of this algorithm for constructing new (to our knowledge) models in anisotropic elasticity theory.
Clebsch-Gordan coefficients of SU(2) and SU(1,1) are defined as eigenfunctions of a linear operator acting on the tensor product of the Hilbert spaces for two irreps of these groups. The shifted harmonic approximation is then used to solve these equations in asymptotic limits in which these eigenfunctions approach harmonic oscillator wave functions and thereby derive asymptotic expressions for these Clebsch--Gordan coefficients.
Generating functions for Clebsch-Gordan coefficients of osp(1|2) are derived. These coefficients are expressed as q goes to - 1 limits of the dual q-Hahn polynomials. The generating functions are obtained using two different approaches respectively based on the coherent-state representation and the position representation of osp(1j2).
Analytic expressions for the Clebsch-Gordan (CG) coefficients of the SO(5) group that involve the 14-dimensional representation can be found in an old paper of M. K. F. Wong. A careful analysis yields that roughly 30% of the coefficients given in that paper are wrong. The correct analytic expressions for all SO(5) group CG coefficients containing the 14-dimensional representation are obtained.
We present sum rules for Clebsch-Gordan coefficients in the framework of SO(4) group-theoretical description of the hydrogen atom. The main results are obtained using properties of the Runge-Lenz- Pauli vector, in particular expressing the matrix elements of the powers of its last component both in spherical and parabolic basis. Connections with Stark effect and diamagnetism of the hydrogen atom are outlined.
The Löwdin-Shapiro projection operator for the Higgs algebra is constructed and utilised to find an analytical expression for the Clebsch-Gordan coefficients for the same.
A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m|n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore, Clebsch-Gordan coefficients corresponding to the tensor product of any covariant tensor representation of gl(m|n) with the natural representation V ([1,0,...,0]) of gl(m|n) with highest weight (1,0,. . . ,0) are computed. Both results are steps for the explicit construction of the parastatistics Fock space.
Clebsch-Gordan coefficients, hypergeometric functions and the binomial distribution
本文证明退化Clebsch-Gordan系数特例可通过二项式系数表示，并揭示其与超几何函数及量子角动量理论中概率概念的内在联系。
We re-express the quantum Calogero-Sutherland model for the Lie algebra E7 and the particular value of the coupling constant K=1 by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra.
We present a new sum rule for Clebsch-Gordan coefficients using generalized characters of irreducible representations of the rotation group. The identity is obtained from an integral involving Gegenbauer ultraspherical polynomials. A similar procedure can be applied for other types of integrals of such polynomials, and may therefore lead to the derivation of further new relations.
Asymptotic expressions for Clebsch-Gordan coefficients are derived from an exact integral representation. Both the classically allowed and forbidden regions are analyzed. Higher-order approximations are calculated. These give, for example, six digit accuracy when the quantum numbers are in the hundreds.
A class of Clebsch-Gordan coefficients are derived from the properties of conditional probability using the binomial distribution. In particular, in the case of $l=l_1+l_2$ it is shown that $$[<l_1/2-k_1, l_2/2-k_2|l/2, k=k_1+k_2]>^2 =\frac{(\begin{array}{c} l_1 k_1\end{array}) (\begin{array}{c}l_2 k_2\end{array})}{(\begin{array}{c}l k \end{array})}$$
Two descriptions of the dual $-1$ Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra $\mathfrak{o}(4)$ and the Lie superalgebra $\mathfrak{osp}(1|2)$.
A new algorithm for computing branching rules and Clebsch-Gordan coefficients of unitary representations of compact groups
本文提出一种受量子力学启发的数值算法，无需群结构代数洞察即可分解紧李群的有限维酉可约表示，并可用于计算Clebsch-Gordan系数。
Tables of scalar factors are presented for 63x63 and 120x63 in SU(8)$\supset$SU(4)xSU(2), and for 35x35 and 56x35 in SU(6)$\supset$SU(3)xSU(2). Related tables for SU(4)$\supset$SU(3)xU(1) and SU(3)$\supset$SU(2)xU(1) are also provided so that the Clebsch-Gordan coefficients can be completely reconstructed. These are suitable to study meson-meson and baryon-meson within a spin-flavor symmetric scheme.
We re-express the quantum Calogero-Sutherland model for the Lie algebra $E_6$ and the particular value of the coupling constant $κ=1$ by using the fundamental irreducible characters of the algebra as dynamical variables. For that, we need to develop a systematic procedure to obtain all the Clebsch-Gordan series required to perform the change of variables. We describe how the resulting quantum Hamiltonian operator can be used to compute more characters and Clebsch-Gordan series for this exceptional algebra.
The paper contains the derivation of a general set of recurrence formulas for the calculus of the SU(3) Clebsch-Gordan coefficients. The first six sections are introductory, presenting the notations and placing SU(3) in the framework of the general group theory. The following eight sections are devoted to the detailed treatment of the carrier spaces of the irreducible representations and their direct product.
We calculate (q-deformed) Clebsch-Gordan and 6j-coefficients for rank two quantum groups. We explain in detail how such calculations are done, which should allow the reader to perform similar calculations in other cases. Moreover, we tabulate the q-Clebsch-Gordan and 6j-coefficients explicitly, as well as some other topological data associated with theories corresponding to rank-two quantum groups. Finally, we collect some useful properties of the fusion rules of particular conformal field theories.
We develop further properties of the matrices $M(m, n, k)$ defined by the author and W. G. Kim in a previous work. In particular, we continue an alternative approach to the theory of Clebsch-Gordan coefficients in terms of combinatorics and convex geometry. New features include a censorship rule for zeros, a sequence of 36-pointed stars of zeros, and another proof of Dixon's Identity. As a major application, we reinterpret the work of Raynal {\it et al.} on vanishing Clebsch-Gordan coefficients as a "middle-out" approach to computing $M(m, n, k).$
Polynomials of the Askey Scheme as Clebsch-Gordan Coefficients
本文证明 Askey 方案中任何有限维多项式族都可解释为某个代数的 Clebsch-Gordan 系数，具体给出了 Hahn、Krawtchouk、Racah 及其 q-变形多项式与振荡子代数、sl_2 及其 q-变形的对应关系。
We discuss the construction and symmetries of su(3) Clebsch-Gordan coefficients arising from the su(3) basis states constructed as triple tensor products of two-dimensional harmonic oscillator states. Because of the su(2) symmetry of the basis states, matrix elements and recursion relations are easily expressed in terms of su(2) technology. As the Weyl group has a particularly simple action on these states, Weyl symmetries of the su(3) coupling coefficients generalizing the well known m-> -m symmetry of su(2) coupling can be obtained, so that any coefficient can be obtained as a sum of Weyl-re…
We present the projection operator method in combination with the Wigner-Racah calculus of the subalgebra U_q(su(2)) for calculation of Clebsch-Gordan coefficients (CGCs) of the quantum algebra U_q(su(3)). The key formulas of the method are couplings of the tensor and projection operators and also a tensor form for the projection operator of U_q(su(3)). We obtain a very compact general analytical formula for the U_q(su(3)) CGCs in terms of the U_q(su(2)) Wigner 3nj-symbols.
We present an algorithm for the explicit numerical calculation of SU(N) and SL(N,C) Clebsch-Gordan coefficients, based on the Gelfand-Tsetlin pattern calculus. Our algorithm is well-suited for numerical implementation; we include a computer code in an appendix. Our exposition presumes only familiarity with the representation theory of SU(2).
Induced representations of Brauer algebra $D_{f}(n)$ from $S_{f_{1}}\times S_{f_{2}}$ with $f_{1}+f_{2}=f$ are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients (ORCs) of $S_{f_{1}}\times S_{f_{2}}\uparrow D_{f}(n)$ with $f\leq 4$ up to a normalization factor are derived by using the linear equation method. Weyl tableaus for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining CG coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebra are proposed. Some iso…
The algebra H of the dual -1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl_{-1}(2). The dual -1 Hahn polynomials are the bispectral polynomials of a discrete argument obtained from a q-> -1 limit of the dual q-Hahn polynomials. The Hopf algebra sl_{-1}(2) has four generators including an involution, it is also a q-> -1 limit of the quantum algebra sl_{q}(2) and furthermore, the dynamical algebra of the parabose oscillator. The algebra H, a two-parameter generalization of u(2) with an involution as additional generator, is first derived from the recurrence r…
We present a program that allows for the computation of tensor products of irreducible representations of Lie algebras A-G based on the explicit construction of weight states. This straightforward approach (which is slower and more memory-consumptive than the standard methods to just calculate dimensions of the tensor product decomposition) produces Clebsch-Gordan coefficients that are of interest for instance in discussing symmetry breaking in model building for grand unified theories. For that purpose, multiple tensor products have been implemented as well as means for analyzing the resultin…
We review the properties of the Kronecker (direct, or tensor) product of square matrices $A \otimes B \otimes C \cdots$ in terms of Hubbard operators. In its simplest form, a Hubbard operator $X_n^{i,j}$ can be expressed as the $n$-square matrix which has entry 1 in position $(i,j)$ and zero in all other entries. The algebra and group properties of the observables that define a multipartite quantum system are notably straightforward in such a framework. In particular, we use the Kronecker product in Hubbard notation to get the Clebsch-Gordan decomposition of the product group $SU(2) \times SU(…
We express each Clebsch-Gordan (CG) coefficient of a discrete group as a product of a CG coefficient of its subgroup and a factor, which we call an embedding factor. With an appropriate definition, such factors are fixed up to phase ambiguities. Particularly, they are invariant under basis transformations of irreducible representations of both the group and its subgroup. We then impose on the embedding factors constraints, which relate them to their counterparts under complex conjugate and therefore restrict the phases of embedding factors. In some cases, the phase ambiguities are reduced to s…
In the present paper we review the $q$-analogue of the Quantum Theory of Angular Momentum based on the $q$-algebra $su_q(2)$, with a special emphasis on the representation of the Clebsch-Gordan coefficients in terms of $q$-hypergeometric series. This representation allows us to obtain several known properties of the Clebsch-Gordan coefficients in an unified and simple way.
Representations of the quantum superalgebra U_q[osp(1/2)] and their relations to the basic hypergeometric functions are investigated. We first establish Clebsch-Gordan decomposition for the superalgebra U_q[osp(1/2)] in which the representations having no classical counterparts are incorporated. Formulae for these Clebsch-Gordan coefficients are derived, and it is observed that they may be expressed in terms of the $Q$-Hahn polynomials. We next investigate representations of the quantum supergroup OSp_q(1/2) which are not well-defined in the classical limit. Employing the universal T-matrix, t…
Representations of a quantum-deformed Lorentz algebra, Clebsch-Gordan map, and Fenchel-Nielsen representation of complex Chern-Simons theory at level-${N}$
本文为量子变形洛伦兹代数构造了一族无限维不可约*-表示，证明其张量积的Clebsch-Gordan分解等价于四孔球面上复Chern-Simons理论中Fenchel-Nielsen长度算符的对角化，并显式给出了相应的谱分解与直积分表示。
A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
本文利用极化迹研究U(sl(N))中心化子，并聚焦sl(3)的L=2情形，构造具有三个生成元的Calabi-Yau代数，证明其PBW基与中心元，揭示Z_2(sl(3))为其商代数，并显式展现E_6型Weyl群对称性及其与Clebsch-Gordan级数的联系。
On the quantum algebra $su_q(1,1)$ from a Special Function standpoint
本文从特殊函数角度研究量子代数 $su_q(1,1)$ 的 Clebsch-Gordan 系数，利用 von Neumann 投影算子方法将其显式表示为对称 q-超几何级数，并统一推导出若干新性质。
In the previous paper math-ph/0507015 we have studied the characters and Clebsch-Gordan series for the exceptional Lie algebra E7 by relating them to the quantum trigonometric Calogero-Sutherland Hamiltonian with coupling constant K=1. Now we extend that approach to the case of general K.
The tensor product of vector and arbitrary representations of the nonstandard q-deformation U'_q(so(n)) of the universal enveloping algebra U(so(n)) of Lie algebra so(n) is defined. The Clebsch-Gordan coefficients of tensor product of vector and arbitrary classical or nonclassical type representations of q-algebra U'_q(so(n)) are found in an explicit form. The Wigner-Eckart theorem for vector operators is proved.
An explicit expression for the general bivariate Krawtchouk polynomials is obtained in terms of the standard Krawtchouk and dual Hahn polynomials. The bivariate Krawtchouk polynomials occur as matrix elements of the unitary reducible representations of SO(3) on the energy eigenspaces of the 3-dimensional isotropic harmonic oscillator and the explicit formula is obtained from the decomposition of these representations into their irreducible components. The decomposition entails expanding the Cartesian basis states in the spherical bases that span irreducible SO(3) representations. The overlap c…
We show that the counting of observables and correlators for a 3-index tensor model are organized by the structure of a family of permutation centralizer algebras. These algebras are shown to be semi-simple and their Wedderburn-Artin decompositions into matrix blocks are given in terms of Clebsch-Gordan coefficients of symmetric groups. The matrix basis for the algebras also gives an orthogonal basis for the tensor observables which diagonalizes the Gaussian two-point functions. The centres of the algebras are associated with correlators which are expressible in terms of Kronecker coefficients…
We study the SL(2,C) Clebsch-Gordan coefficients appearing in the lorentzian EPRL spin foam amplitudes for loop quantum gravity. We show how the amplitudes decompose into SU(2) nj-symbols at the vertices and integrals over boosts at the edges. The integrals define edge amplitudes that can be evaluated analytically using and adapting results in the literature, leading to a pure state sum model formulation. This procedure introduces virtual representations which, in a manner reminiscent to virtual momenta in Feynman amplitudes, are off-shell of the simplicity constraints present in the theory, b…
We determine the Clebsch-Gordan and Racah-Wigner coefficients for continuous series of representations of the quantum deformed algebras U_q(sl(2)) and U_q(osp(1|2)). While our results for the former algebra reproduce formulas by Ponsot and Teschner, the expressions for the orthosymplectic algebra are new. Up to some normalization factors, the associated Racah-Wigner coefficients are shown to agree with the fusing matrix in the Neveu-Schwarz sector of N=1 supersymmetric Liouville field theory.
It is well known that there exist non-isomorphic compact groups with isomorphic representation rings (fusion rules). Nevertheless, considerable structural information about the group can be reconstructed from its representation ring. We review these types of partial reconstruction theorems, including some recent results. In the Appendix a derivation of the Clebsch-Gordan series of SU(2) based only on information about the dimensions of the irreps is presented.
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