Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Dunkl oscillator in the plane II : representations of the symmetry
  algebra   The superintegrability, wavefunctions and overlap coefficients of the Dunkl
oscillator model in the plane were considered in the first part. Here
finite-dimensional representations of the symmetry algebra of the system,
called the Schwinger-Dunkl algebra sd(2), are investigated. The algebra sd(2)
has six generators, including two involutions and a central element, and can be
seen as a deformation of the Lie algebra u(2). Two of the symmetry generators,
J_3 and J_2, are respectively associated to the separation of variables in
Cartesian and polar coordinates. Using the parabosonic creation/annihilation
operators, two bases for the representations of sd(2), the Cartesian and
circular bases, are constructed. In the Cartesian basis, the operator J_3 is
diagonal and the operator J_2 acts in a tridiagonal fashion. In the circular
basis, the operator J_2 is block upper-triangular with all blocks 2x2 and the
operator J_3 acts in a tridiagonal fashion. The expansion coefficients between
the two bases are given by the Krawtchouk polynomials. In the general case, the
eigenvectors of J_2 in the circular basis are generated by the Heun polynomials
and their components are expressed in terms of the para-Krawtchouk polynomials.
In the fully isotropic case, the eigenvectors of J_2 are generated by little -1
Jacobi or ordinary Jacobi polynomials. The basis in which the operator J_2 is
diagonal is then considered. In this basis, the defining relations of the
Schwinger-Dunkl algebra imply that J_3 acts in a block tridiagonal fashion with
all blocks 2x2. The matrix elements of J_3 in this basis are given explicitly.

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The categories of this paper are: q-alg
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The categories of this paper are: physics
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The categories of this paper are: econ
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The categories of this paper are: quant-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: acc-phys