Abstract. The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is superintegrable and its symmetry generators are obtained by the Schwinger construction using parabosonic creation/annihilation operators. The algebra generated by the constants of motion, which we term the Schwinger-Dunkl algebra, is an extension of the Lie algebra u(2) with involutions. The system admits separation of variables in both Cartesian and polar coordinates. The separated wavefunctions are respectively expressed in terms of generalized Hermite polynomials and products of Jacobi and Laguerre polynomials. Moreover, the so-called Jacobi-Dunkl polynomials appear as eigenfunctions of the symmetry operator responsible for the separation of variables in polar coordinates. The expansion coefficients between the Cartesian and polar bases (overlap coefficients) are given as linear combinations of dual −1 Hahn polynomials. The connection with the Clebsch-Gordan problem of the sl −1 (2) algebra is explained.The Dunkl oscillator in the plane I 2

The Dirac-Dunkl operator on the 2-sphere associated to the Z 3 2 reflection group is considered. Its symmetries are found and are shown to generate the Bannai-Ito algebra. Representations of the Bannai-Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac-Dunkl operator are obtained using a Cauchy-Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai-Ito algebra.

Abstract. The Bannai-Ito polynomials are shown to arise as Racah coefficients for sl −1 (2). This Hopf algebra has four generators including an involution and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for sl −1 (2). The Racah coefficients are recovered from a related Leonard pair.

We study 1+1 dimensional φ 4 theory using the recently proposed method of conformal truncation. Starting in the UV CFT of free field theory, we construct a complete basis of states with definite conformal Casimir, C. We use these states to express the Hamiltonian of the full interacting theory in lightcone quantization. After truncating to states with C ≤ C max , we numerically diagonalize the Hamiltonian at strong coupling and study the resulting IR dynamics. We compute non-perturbative spectral densities of several local operators, which are equivalent to real-time, infinite-volume correlation functions. These spectral densities, which include the Zamolodchikov C-function along the full RG flow, are calculable at any value of the coupling. Near criticality, our numerical results reproduce correlation functions in the 2D Ising model.

Abstract. The kernel of the Z n 2 Dirac-Dunkl operator is examined. The symmetry algebra An of the associated Dirac-Dunkl equation on S n−1 is determined and is seen to correspond to a higher rank generalization of the Bannai-Ito algebra. A basis for the polynomial null-solutions of the DiracDunkl operator is constructed. The basis elements are joint eigenfunctions of a maximal commutative subalgebra of An and are given explicitly in terms of Jacobi polynomials. The symmetry algebra is shown to act irreducibly on this basis via raising/lowering operators. A scalar realization of An is proposed.

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere, is cast in the framework of the Racah problem for the su(1, 1) algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three su(1, 1) algebras, respectively. The construction makes explicit the isomorphism between the Racah-Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for su (1, 1), and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah-Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace-Dunkl operator associated to the Z n 2 root system. This algebra is also the invariance algebra of the generic superintegrable model on the n-sphere. Bases of Dunkl harmonics are constructed explicitly using a Cauchy-Kovalevskaia theorem. These bases consist of joint eigenfunctions of labelling Abelian subalgebras of the higher rank Racah algebra. A method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented.A HIGHER RANK RACAH ALGEBRA AND THE Z n 2 LAPLACE-DUNKL OPERATOR 3

The universal character of the Racah algebra will be illustrated by showing that it is at the center of the relations between the Racah polynomials, the recoupling of three su(1, 1) representations and the symmetries of the generic second-order superintegrable model on the 2-sphere.