Finite-dimensional representations of the symmetry algebra of the dihedral Dunkl–Dirac operator

The centre of the Dunkl total angular momentum algebra

“…In [11] (see also [31]) the case of sl(2) was considered and results on the joint-decomposition of the polynomial space for the action of the dual pair was obtained. Similar results for osp(1|2) were obtained in [15], while in [29] the symmetry algebra was completely determined (see also [28] for an initial overview of the theory of deformations of the Howe dual pairs we consider in the Cherednik context, [12] for the case when W is a product of groups of type A 1 , which is used to define a higher rank Bannai-Ito algebra, and [16], [13] for other specific choices of the reflection group W ).…”

The Dihedral Dunkl–Dirac Symmetry Algebra with Negative Clifford Signature

“…Doing it would, however, greatly exceed the scope of this note. We give below a sketch of the steps needed and refer the readers to [2] for the details.…”

“…In a recent article [2], we constructed the finite-dimensional representations of the dihedral Dunkl-Dirac symmetry algebra SA m ⊂ A κ ⊗ Cl (3), that is, the symmetry algebra of the osp(1|2)-realisation linked to the group W = Z 2 × D 2m acting on the three-dimensional Euclidean space. A pair of ladder operators behaving nicely under the action of the double covering W of the group W was instrumental to this.…”

“…We study thus here the algebra SA ε m ⊂ A κ ⊗ Cl ε (3). The complete classification of the finite-dimensional representations is long and would greatly exceed the allowed space, we refer the readers to [2] for its details. We believe this contribution could be of help for interested readers who want to translate our results, since both conventions for Clifford algebras coexist and the two lead to non-isomorphic real Clifford algebras; multiplication of the generators by i gives the correspondence for complex Clifford algebras.…”

“…Any finite-dimensional SA ε m -representation decomposes as a W ε -representation into a direct sum of spin irreducible W ε -representations by Maschke's Theorem (the irreducible representations for these groups can be found in[2, Thm A.5]). Let W ε 0 be the subgroup of W ε generated by elements commuting with O 0 .…”