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We consider the symmetry algebra generated by the total angular momentumoperators, appearing as constants of motion of the $\mathrm{S}_3$ Dunkl Diracequation. The latter is a deformation of the Dirac equation by means of Dunkloperators, in our case associated to the root system $A_2$, with correspondingWeyl group $\mathrm{S}_3$, the symmetric group on three elements. The explicitform of the symmetry algebra in this case is a one-parameter deformation of theclassical total angular momentum algebra $\mathfrak{so}(3)$, incorporatingelements of $\mathrm{S}_3$. This was obtained using recent results on thesymmetry algebra for a class of Dirac operators, containing in particular theDirac-Dunkl operator for arbitrary root system. For this symmetry algebra, weclassify all finite-dimensional, irreducible representations and determine theconditions for the representations to be unitarizable. The class of unitaryirreducible representations admits a natural realization acting on arepresentation space of eigenfunctions of the Dirac Hamiltonian. Using aCauchy-Kowalevsky extension theorem we obtain explicit expressions for theseeigenfunctions in terms of Jacobi polynomials.

The total angular momentum algebra related to the $\mathrm{S}_3$ Dunkl Dirac equation