Related papers: On a general analytical formula for U_q(su(3))-Cle…
We use the eigenfunction method to calculate the Clebsh-Gordan coefficients for the permutation group . This method is well-established by Jin-Quan Chen. Here we elaborate the detailed procedures for the pedagogical purpose. Due to the…
A 'differential measure' is used to cast our calculus for the group $SU(3)$ into a form similar to Schwinger's boson operator calculus for the group $SU(2)$. It is then applied to compute (i) the inner product between the basis states and…
A theory of Clebsch-Gordan coefficients for $SL(2, C)$ is given using only rational numbers. Features include orthogonality relations, recurrence relations, and Regge's symmetry group. Results follow from elementary representation theory…
We give a fully explicit description of Lie algebra derivatives (generalizing raising and lowering operators) for representations of SL(3,R) in terms of a basis of Wigner functions. This basis is natural from the point of view of principal…
$GL_h(n) \times GL_h(m)$-covariant $h$-bosonic algebras are built by contracting the $GL_q(n) \times GL_q(m)$-covariant $q$-bosonic algebras considered by the present author some years ago. Their defining relations are written in terms of…
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)…
The Clebsch-Gordan coefficients or Wigner $3j$ symbols are known to be proportional to a $_3F_2(1)$ hypergeometric series, and Racah $6j$ coefficients to a $_4F_3(1)$. In general, however, non-trivial $9j$ symbols can not be expressed as a…
Fast, accurate, and stable computation of the Clebsch-Gordan (C-G) coefficients is always desirable, for example, in light scattering simulations, the translation of the multipole fields, quantum physics and chemistry. Current recursive…
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 a semisimple compact Lie group $G$ and a nonzero dominant integral weight $\lambda$, the highest weight $G_q$-modules $V_{n\lambda}$ form a subproduct system of finite dimensional Hilbert spaces. Using a conjectural asymptotic…
In this paper, we give a generic algorithm of the transition operators between Hermitian Young projection operators corresponding to equivalent irreducible representations of SU(N), using the compact expressions of Hermitian Young…
The quantum Clebsch-Gordan coefficients and the explicit form of the $\breve{R}_{q}$ matrix related with the minimal representation of the quantum enveloping algebra $U_{q}E_{7}$ are calculated in this paper.
We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a…
A special case of Askey-Wilson algebra $AW(3)$ with three generators is shown to serve as a hidden symmetry algebra underlying the Hahn problem for the quantum algebra $sl_q(2)$. On the base of this hidden symmetry the corresponding…
We introduce a Dirac operator $D$ for the quantum group $U_q(\mathfrak{sl}_2)$, as an element of the tensor product of $U_q(\mathfrak{sl}_2)$ with the Clifford algebra on two generators. We study the properties of $D$, including an analogue…
A general class of W-algebras can be constructed from the affine sl(N) algebra by (quantum) Drinfeld-Sokolov reduction and are classified by partitions of N. Surface operators in an N=2 SU(N) 4d gauge theory are also classified by…
We will derive both quaternion and octonion algebras as the Clebsch-Gordan algebras based upon the su(2) Lie algebra by considering angular momentum spaces of spin one and three. If we consider both spin 1 and 1/2 states, then the same…
We consider intertwining relations of the triangular $q$-Onsager algebra, and obtain generic triangular boundary $K$-operators in terms of the Borel subalgebras of $U_{q}(sl_2)$. These $K$-operators solve the reflection equation.
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…
Given a semi-simple algebra equipped with a coproduct, the Clebsch--Gordan coefficients are the elements of the transition matrices between direct product representation and its irreducible decomposition. It is well known that the…