Coefficients Of SU Research Articles

3nj-coefficients of su(1,1) as connection coefficients between orthogonal polynomials in n variables

In the tensor product of n+1 positive discrete series representations of su(1,1), a coupled basis vector can be described by a certain binary coupling tree. To every such binary coupling tree, polynomials Rl(k)(x) and Rl(k)(x) are associated. These polynomials are n-variable Jacobi and continuous Hahn polynomials, and are orthogonal with respect to a weight function. The connection coefficients expressing such a polynomial associated with a given binary coupling tree in terms of those polynomials associated with another binary coupling tree are proportional to 3nj-coefficients of su(1,1).

The multiple sum formulas for 12j coefficients of SU(2) and uq(2)

The expressions for 12j coefficients of the both kinds (without and with braiding) of the SU(2) group and the quantum algebra uq(2) are considered. Using Dougall’s summation formula of the very well-poised hypergeometric F45(1) series and its q-generalization, several fourfold sum formulas [with each sum related to the balanced F45(1) or φ45 series] for the q-12j coefficients of the second kind (without braiding) are derived. Applying q-generalizations of rearrangement formulas of the very well-poised hypergeometric F56(−1) series [which correspond to a new expression for the Clebsch–Gordan coefficients of SU(2) and uq(2)], the new expressions with five sums [of the F34(1) and F23(1) or φ45 and φ23 type] are derived for the q-12j coefficients of the first kind (with braiding) instead of the usual expansions in terms of q-6j coefficients. Stretched and doubly stretched q-12j coefficients [as triple, double, or single sums, related to composed or separate hypergeometric F34(1) and F45(1) or φ33 and φ45 series and, particularly, to q-9j or q-6j coefficients] are considered.

Transformation formulas for double hypergeometric series related to 9-j coefficients and their basic analogs

In a recent paper, Ališauskas deduced different triple sum expressions for the 9-j coefficient of su(2) and suq(2). For a singly stretched 9-j coefficient, these reduce to different double sum series. Using these distinct series, we deduce a set of new transformation formulas for double hypergeometric series of Kampé de Fériet type and their basic analogs. These transformation formulas are valid for rather general parameters of the series, although a common feature is that all the series appearing here are terminating. It is also shown that the transformation formulas deduced here generate a group of transformation formulas, thus yielding an invariance group or symmetry group of particular double series.

Hypergeometric series related to the 9- j coefficient of [formula omitted

We study various expressions for the 9- j coefficient of su(1,1) , with an emphasis on multiple hypergeometric series. An orthogonal polynomial in two discrete variables is associated to the 9- j coefficient. Some applications are considered, dealing with formulas in which the 9- j coefficient appears as a connection or expansion coefficient, and discussing some transformation formulas for double- and triple-hypergeometric series of unit argument.

The hidden angular momenta of the coupling-recoupling coefficients ofSU(2)

A finite projective geometry T = PG (n ,2) is associated with any coupling-recoupling (N -jm ) coefficient of SU (2). This geometry is based on a duality of projective spaces and a discrete Fourier transform. An angular momentum jk with projection Mk is attached at each point k T . Some of these momenta and projections are specified by the arguments of the N -jm coefficient. The others are qualified as hidden. The value of the N -jm coefficient is given in terms of a summation over the hidden angular momenta and hidden projections of a `full pn -JM symbol' with a high degree of symmetry. For the 3-j coefficient (Clebsch-Gordan or Wigner coefficient), the finite projective geometry is a line of three points with one hidden projection and the formula of hidden momenta gives an interpretation of the combinatorial formula of Racah for the 3-j coefficient.

The representations and coupling coefficients of su(n); application to su(4)

Analytical expressions for the matrices and an explicit algorithm for computing Clebsch-Gordan coupling coefficients are given forsu(4) in au(3)-coupled basis as an example of the construction for anysu(n) in au(n−1) basis. The results areinduced from the known results foru(3) by means of the vector-coherent-state (VCS) theory of induced representations. The important recent result that makes this possible is the discovery that a complete set of shift tensors for the finitedimensional representations of reductive Lie algebras can be induced, by VCS methods, from those of suitably defined subalgebras.

Integration over the group space of SUq(2), Clebsch-Gordan coefficients, and related identities

We calculate the Clebsch-Gordan coefficients of SUq(2) by a Woronowicz integration over the group manifold and obtain a representation differing from that reached by working with theq-group algebra. These apparently different results must agree, however, and their equivalence implies aq-identity. On lettingq = 1, we shall obtain two results of different structures for the Clebsch-Gordan coefficients of SU(2) and their equivalence similarly implies an identity among the usual binomial coefficients. With the same approach, one may extend the Woronowicz integral of the product of two irreducible representations to products of many irreducible representations.

On the zeros of 6-j coefficients

An extensive computer search for non-trivial zeros of the 6-j coefficients of SU(2) has been performed to find their frequency as a function of the sum of their quantum numbers and of their polynomial degree. For the zeros of degree 2, four of the seven cases already known (taking into account Regge symmetries) of two relations between quantum numbers relevant to a Pell equation have been generalized to deal with only one relation; a new case of two relations and many new cases of three relations, all with polynomial solutions. Have been found; but all these formulae only account for one zero out of seven. For degree 3, we found four new cases of two relations and some other cases of three relations with polynomial solutions; with the already known case of a Pell equation, they account for almost half of the zeros. The fast decrease of the number of zeros with the degree and the relative smallness of their quantum number is an indication that there should be no zeros of degree larger than 9.

Q-TENSOR METHOD OF DETERMINING IRREDUCIBLE REPRESENTATION OF QUANTUM ALGEBRA SUq(3)

In this paper, we construct irreducible q-tensor operators of rank [Formula: see text] of quantum algebra SU q(2) using the generators of quantum algebra SU q(3). By means of property of q-tensor operator, we can easily obtain irreducible representation (IR) of SU q(3). A recurrent formula to calculate reduced coefficients of SU q(3) is obtained and the multiplicity of reduced coefficients is discussed.

Recursive calculation of the R matrices of q-deformed algebras

The authors show that the pentagonal equation, relating R-matrices and vector coupling coefficients, can be used to calculate recursively general R-matrices from those containing the primitive irrep. All R-matrices for su(2)q are calculated by this method. As an illustration of the more general case the primitive vector coupling coefficients for su(3)q are obtained and used to derive formulae for the R-matrices with a certain restriction on the representations.

Explicit orthonormal Clebsch–Gordan coefficients of SU(3)

The construction of the explicit algebraic-polynomial expressions for the nonmultiplicity-free orthonormal Clebsch–Gordan (Wigner) coefficients of SU(3)⊇U(2) is completed in the case of the paracanonical coupling scheme related with the explicit minimal biorthogonal systems by means of the Hecht or Gram–Schmidt process. The direct and inverse orthogonalization coefficients (the first of them being equivalent to the boundary orthonormal isofactors) are expressed, up to explicitly given multiplicative factors, in terms of the numerator and denominator polynomials related with the auxiliary Aλ function of Louck, Biedenharn, and Lohe that appears as a fragment of the denominator G-functions of canonical SU(3) tensor operators.

Construction of coupling coefficients of SU(4) in a supermultiplet basis

Matrix elements of the unit tensor operators of ranks (1), (2) and (1,1) in the non-orthonormal Draayer basis of SU(4) are found. The evaluation algorithms for the SU(4) coupling coefficients are proposed for coupling one arbitrary and one four-, six- or ten-dimensional irreducible representation of SU(4) in the SU(2)*SU(2) chain, as well as in the case of the semistretched coupling. The construction of the overlaps of the Draayer basis states is considered.

A resolution of the SU(3) outer multiplicity problem and computation of Wigner coefficients for SU(3)

A simple algorithm is given for the resolution of the SU(3) multiplicity problem and the computation of SU(3) Wigner coefficients using a complete set of U(2)(X)U(3) Bargmann tensors classified by operator patterns. Null space properties of these tensors are easily derived. Their structure is such that a direct one-to-one correspondence is shown to exist between the Bargmann tensors and the terms in the Clebsch-Gordan series for SU(3) as derived by O'Reilly (1982). Finally, the resolution presented herein is shown to be concordant with an alternative resolution advocated long ago by Hecht.

Comment on 'New relations between the Clebsch-Gordan coefficients of SU(2)'

The relations investigated by J. Kulesza and J. Rembielinski (see ibid., vol.13, p.1189-95, 1980) are particular cases of a recoupling equation.

New relations between the Clebsch-Gordan coefficients of SU(2)

Some unknown bilinear relations between the SU(2) Clebsch-Gordan coefficients are given. The derivation is based on the analysis of the triplet P contains/implies L contains/implies SO(3), (Poincare, Lorentz and rotation group). The form of the covariant pmu -dependent functions with fixed maximal degree with respect to pmu is given.

Spin-zero mesons and current algebras

Large chiral algebras, using the $f$ and $d$ coefficients of SU(3) can be constructed with spin-$\frac{1}{2}$ baryons. Such algebras have been found useful in some previous investigations. This article examines under what conditions similar or identical current algebras may be realized with spin-0 mesons. A curious lack of analogy emerges between meson and baryon currents. Second-class currents, made of mesons, are required in some algebras. If meson and baryon currents are to satisfy the same extended SU(3) algebra, four meson nonets are needed, in terms of which we give an explicit construction for the currents.

Spin recoupling and n-electron matrix elements

For n-electron systems with well defined total spin antisymmetric states are constructed by successively coupling the spins associated with each orbital. A second quantized scheme is used and the matrix elements of these states are expressed both for spin-independent and spin-dependent interactions in terms of recoupling coefficients of SU(2). The latter are evaluated to give very simple expressions. As a particular case a simple formula for the matrix elements of generators of U(N) for two-column partitions is obtained.

Mixed-Basis D Functions and Clebsch-Gordan Coefficients of Noncompact Groups

Mixed-basis D functions are introduced as a tool for deriving Clebsch-Gordan coefficients of induced representations of semisimple groups. The Clebsch-Gordan coefficients of SU(1, 1) and SL(2, C) are computed as examples.

Note on a Class of CG Coefficients ofSU(3)

Note on a Class of CG Coefficients of<i>S</i><i>U</i>(3)

On the representations of the group SU(3)

By a generalization of techniques developed earlier for dealing with the three-dimensional rotation group O3 the generators of the group SU(3) are expressed as differential operators involving four independent variables. The reduction in the number of variables simplifies the mathematical problem and makes it easier to study the properties of the group and its irreducible representations. From the forms of the new operators it becomes apparent that the basic states of an irreducible representation of SU(3) are linear combinations of Clebsch-Gordan series of SU(2) (or O3). A convenient expression for the latter, which simplifies the derivation and also brings out the significance of certain steps more clearly, is obtained here by a proper interpretation of the results given in an earlier paper by the author. Besides this certain recursion relations for the Clebsch-Gordan coefficients of SU(2) are also found to be helpful for studying the SU(3) representations. These relations are derived in a novel way from Gauss's relations between contiguous hypergeometric functions.