Some time ago, the generalization of ordinary spherical harmonics to spinweighted functions (e.g., [33, 28, 56]) was developed to allow for harmonic expansions of spinweighted functions on the sphere. In this paper we have instead used the tensorial form of these spinweighted harmonics, the tensorial spins spherical harmonics, which are formed by taking appropriate linear combinations of the [59]:

where the indices obey , and the number of spatial indices (i.e., ) is equal to . Explicitly, these tensorial spinweighted harmonics can be constructed directly from the parametrized Lorentzian null tetrad, Eq. (1.1) – (1.2):
Taking the spatial parts of their duals, we obtain the oneforms From this we define as [59] The other harmonics are determined by the action of the operator on the forms, Eq. (C.2), (with complex conjugates) via Specifically, the spin harmonics are defined byWe now present a table of the tensorial spherical harmonics up to , in terms of the tetrad. Higher harmonics can be found in [59].
, 
, 
, , 
, 
In addition, it is useful to give the explicit relations between these different harmonics in terms of the operator and its conjugate. Indeed, we can see generally that applying once raises the spin index by one, and applying lowers the index by one. This in turn means that
Finally, due to the nonlinearity of the theory, we have been forced throughout this review to consider products of the tensorial spin spherical harmonics while expanding nonlinear expressions. These products can be expanded as a linear combination of individual harmonics using Clebsch–Gordon expansions. The explicit expansions for products of harmonics with or are given below (we omit higher products due to the complexity of the expansion expressions). Further products can be found in [59, 42].
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Living Rev. Relativity 15, (2012), 1
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