Related papers: An elementary approach to 6j-symbols (classical, q…
This paper treats 6j symbols or their orthonormal forms as a function of two variables spanning a square manifold which we call the "screen". We show that this approach gives important and interesting insight. This two dimensional…
We relate quantum 6J symbols of various types (quantum versions of Wigner and Racah symbols) to Ocneanu cells associated with AN Dynkin diagrams. We check explicitly the algebraic structure of the associated quantum groupoids and analyze…
We compute some 6j symbols in the category of polynomial representations of $GL(\infty)$.
We present the asymptotic formula for the Wigner 9j-symbol, valid when all quantum numbers are large, in the classically allowed region. As in the Ponzano-Regge formula for the 6j-symbol, the action is expressed in terms of lengths of edges…
We present new asymptotic formulas for the Wigner $15j$-symbol with two, three, or four small quantum numbers, and provide numerical evidence of their validity. These formulas are of the WKB form and are of a similar nature as the…
We establish the geometry behind the quantum $6j$-symbols under only the admissibility conditions as in the definition of the Turaev-Viro invariants of $3$-manifolds. As a classification, we show that the $6$-tuples in the quantum…
The 6j-symbol is a fundamental object from the re-coupling theory of SU(2) representations. In the limit of large angular momenta, its asymptotics is known to be described by the geometry of a tetrahedron with quantized lengths. This…
We conjecture a closed form expression for the simplest class of multiplicity-free quantum 6j-symbols for U_q(sl_N). The expression is a natural generalization of the quantum 6j-symbols for U_q(sl_2) obtained by Kirillov and Reshetikhin.…
We show how a simple and elegant graphical notation can be used to derive the Biedenharn-Elliot identity for the 6j-symbol and we demonstrate how the same technique can be applied to obtain new identities for the 6j. We then employ the same…
New bispectral polynomials orthogonal on a quadratic bi-lattice are obtained from a truncation of Wilson polynomials. Recurrence relation and difference equation are provided. The recurrence coefficients can be encoded in a perturbed…
A well known recurrence relation for the 6j-symbol of the quantum group su_q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking…
We study the asymptotic expansion of the 6j-symbol using the Schulten-Gordon recursion relations. We focus on the particular case of the isosceles tetrahedron and we provide explicit formulas for up to the third order corrections beyond the…
We derive a new asymptotic formula for the Wigner $9j$-symbol, in the limit of one small and eight large angular momenta, using a novel gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our…
The $9j$ symbols of $\mathfrak{su}(1,1)$ are studied within the framework of the generic superintegrable system on the 3-sphere. The canonical bases corresponding to the binary coupling schemes of four $\mathfrak{su}(1,1)$ representations…
It has been known since 2007 that the Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere,…
We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in…
A new uniform asymptotic approximation for the Wigner $6j$ symbol is given in terms of Wigner rotation matrices ($d$-matrices). The approximation is uniform in the sense that it applies for all values of the quantum numbers, even those near…
We study the tensor product of principal unitary representations of the quantum Lorentz group, prove a decomposition theorem and compute the associated intertwiners. We show that these intertwiners can be expressed in terms of complex…
On basis of generalized 6j-symbols we give a formulation of topological quantum field theories for 3-manifolds including observables in the form of coloured graphs. It is shown that the 6j-symbols associated with deformations of the…
It is shown that the well known Racah sum rule and Biedenharn-Elliott identity satisfied by the recoupling coefficients or by the $6-j$ symbols of the usual rotation $SO(3)$ algebra can be extended to the corresponding features of the…