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We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkloperator associated with the Weyl group of the exceptional root system $G_2$.For this symmetry algebra, we give both an abstract definition and an explicitrealisation. We then construct ladder operators, using an intermediate resultwe prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finitereflection group acting on a three-dimensional space.

An exceptional symmetry algebra for the 3D Dirac-Dunkl operator