  We consider the symmetry algebra generated by the total angular momentum
operators, appearing as constants of motion of the $\mathrm{S}_3$ Dunkl Dirac
equation. The latter is a deformation of the Dirac equation by means of Dunkl
operators, in our case associated to the root system $A_2$, with corresponding
Weyl group $\mathrm{S}_3$, the symmetric group on three elements. The explicit
form of the symmetry algebra in this case is a one-parameter deformation of the
classical total angular momentum algebra $\mathfrak{so}(3)$, incorporating
elements of $\mathrm{S}_3$. This was obtained using recent results on the
symmetry algebra for a class of Dirac operators, containing in particular the
Dirac-Dunkl operator for arbitrary root system. For this symmetry algebra, we
classify all finite-dimensional, irreducible representations and determine the
conditions for the representations to be unitarizable. The class of unitary
irreducible representations admits a natural realization acting on a
representation space of eigenfunctions of the Dirac Hamiltonian. Using a
Cauchy-Kowalevsky extension theorem we obtain explicit expressions for these
eigenfunctions in terms of Jacobi polynomials.
