Related papers: Elliptic hypergeometric function and $6j$-symbols …
We classify all pairs $(V,H)$, where $H$ is a proper subgroup of $G=Sp_6(q)$, $q$ even, and $V$ is an $\ell$-modular representation of $G$ for $\ell\neq 2$ which is absolutely irreducible as a representation of $H$. This problem is…
We propose an algebraic expression for $U_q(\mathfrak{sl}_3)$ quantum $3j$ symbols (quantum Clebsch-Gordan coefficients) appearing in the decomposition of tensor product of symmetric representations. Our compact form will be useful to write…
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 is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function ${}_2F_1$ arises from a relation between modular curves, namely the covering of $X_0(3)$ by $X_0(9)$. In general, when $2\le N\le 7$ the N-fold cover of…
We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_2 WZW model of CFT. The modular group acts on this space. We…
We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlev\'e equation (or higher-order analogues), and admitting a large family of monodromy-preserving…
Algebraic hypergeometric functions can be compactly expressed as radical or dihedral functions on pull-back curves where the monodromy group is much simpler. This article considers the classical 3F2-functions with the projective monodromy…
We consider the action of the Lie algebra of polynomial vector fields, $\mathfrak{vect}(1)$, by the Lie derivative on the space of symbols $\mathcal{S}_\delta^n=\bigoplus_{j=0}^n \mathcal{F}_{\delta-j}$. We study deformations of this…
What discuss the problem of obtaining new manifold invariants via different analogues of 6j-symbols and the torsion of acyclic complexes.
Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a…
We define a set of orthogonal functions on the complex projective space CP^{N-1}, and compute their Clebsch-Gordan coefficients as well as a large class of 6-j symbols. We also provide all the needed formulae for the generation of…
For a fixed SL(3, Z) Maass form g, we consider the family of L-functions L(g \times u_j, s) where u_j runs over the family of Hecke-Maass cusp forms on SL(2,Z). We obtain an estimate for the second moment of this family of L-functions at…
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.
In recent work (math.QA/0309252) on multivariate hypergeometric integrals, the author generalized a conjectural integral formula of van Diejen and Spiridonov to a ten parameter integral provably invariant under an action of the Weyl group…
Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang-Baxter equation, which is precisely the Yang-Baxter equation satisfied by 6j-symbols. We investigate one of…
The Kashaev invariants of 3-manifolds are based on $6j$-symbols from the representation theory of the Weyl algebra, a Hopf algebra corresponding to the Borel subalgebra of $U_q(sl(2,\C))$. In this paper, we show that Kashaev's $6j$-symbols…
We discuss our conjecture for simply laced Lie algebras level two string functions of mark one fundamental weights and prove it for the $SO(2r)$ algebra. To prove our conjecture we introduce $q$-diagrams and examine the diagrammatic…
The construction, in [AJN], of a pseudodifferential calculus analogous to the Weyl calculus, in an infinite dimensional setting, required the introduction of convenient classes of symbols. In this article, we proceed with the study of these…
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…
Simultaneous eigenfunctions of two Askey-Wilson second order difference operators are constructed as formal matrix coefficients of the principal series representation of the modular double of the quantized universal enveloping algebra…