Related papers: An elementary approach to 6j-symbols (classical, q…
The semiclassical mechanics of the Wigner 6j-symbol is examined from the standpoint of WKB theory for multidimensional, integrable systems, to explore the geometrical issues surrounding the Ponzano-Regge formula. The relations among the…
We discuss in details the role of Wigner 6j symbol as the basic building block unifying such different fields as state sum models for quantum geometry, topological quantum field theory, statistical lattice models and quantum computing. The…
Upon solving a finite discrete reduction of the difference Heun equation, we arrive at an elliptic generalization of the Racah polynomials. We exhibit the three-term recurrence relation and the orthogonality relations for these elliptic…
A limit formula from q-Racah polynomials to big q-Jacobi polynomials is given which can be considered as a limit formula for orthogonal polynomials. This is extended to a multi-parameter limit with 3 parameters, also involving (q-)Hahn…
These notes contain essentially a rewriting of several properties of two well-known quantities, the so-called theta symbol (or triangular symbol), which is rational, and the 6j symbol, which is usually irrational, in terms of two related…
The relation between Wilson and para-Racah polynomials and representations of the degenerate rational Sklyanin algebra is established. Second order Heun operators on quadratic grids with no diagonal terms are determined. These special or…
It is well known that the building blocks for state sum models of quantum gravity are given by 6j and 10j symbols. In this work we study the asymptotics of these symbols by using their expressions as group integrals. We carefully describe…
The aim of this article is to study the limit transitions from non-standard q-Racah polynomials to big q-Jacobi, dual q-Hahn, and q-Hahn polynomials such that the orthogonality properties and the three-term recurrence relations remain…
The Sklyanin algebra admits realizations by difference operators acting on theta functions. Sklyanin found an invariant metric for the action and conjectured an explicit formula for the corresponding reproducing kernel. We prove this…
The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of $U_q(\mathfrak{sl}_2)$ are considered. It is shown that these intermediate Casimirs are…
The corrected triple sum expression of Ali\v{s}auskas (1987) for the recoupling (Racah) coefficients (6j-symbols) of the symmetric (most degenerate) representations of the orthogonal groups SO(n) (previously derived from the fourfold sum…
In the paper a simple explicit formula for an arbitrary $3j$-symbol for the Lie algebra $\mathfrak{gl}_3$ is given. More precise necessary conditions for non-vanishing of a $3j$-symbol are given, in the case when these conditions hold we…
For q a root of unity of order 2r, we give explicit formulas of a family of 3-variable Laurent polynomials J_{i,j,k} with coefficients in Z[q] that encode the 6j-symbols associated with nilpotent representations of U_qsl_2. For a given…
We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential…
In this paper, we consider an extension of Jacobi's symbol, the so called rational $2^k$-th power residue symbol. In Section 3, we prove a novel generalization of Zolotarev's lemma. In Sections 4, 5 and 6, we show that several hard…
We generalize the colored Alexander invariant of knots to an invariant of graphs, and we construct a face model for this invariant by using the corresponding 6j-symbol, which comes from the non-integral representations of the quantum group…
We present an efficient implementation for the evaluation of Wigner 3j, 6j, and 9j symbols. These represent numerical transformation coefficients that are used in the quantum theory of angular momentum. They can be expressed as sums and…
The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulae given by Edmonds, which relate the 3j and 6j symbols to rotation matrices.
The main subject of the paper is the pentagon relation. This relation can be expressed in different ways. We start with the natural geometric form of the pentagon relation. Then we express it in algebraic form as a family of equations with…
We derive an asymptotic formula for the Wigner $12j$ symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the $12j$ symbol with one small angular momentum. We present the first kind…