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Spin-weighted spherical functions provide a useful tool for analyzingtensor-valued functions on the sphere. A tensor field can be decomposed intocomplex-valued functions by taking contractions with tangent vectors on thesphere and the normal to the sphere. These component functions are usuallypresented as functions on the sphere itself, but this requires an implicitchoice of distinguished tangent vectors with which to contract. Thus, we maymore accurately say that spin-weighted spherical functions are functions ofboth a point on the sphere and a choice of frame in the tangent space at thatpoint. The distinction becomes extremely important when transforming thecoordinates in which these functions are expressed, because the implicit choiceof frame will also transform. Here, it is proposed that spin-weighted sphericalfunctions should be treated as functions on the spin group. This approach morecleanly reflects the geometry involved, and allows for a more elegantdescription of the behavior of spin-weighted functions. In this form, thespin-weighted spherical harmonics have simple expressions as elements of theWigner $\mathfrak{D}$ representations, and transformations under rotation aresimple. Two variants of the angular-momentum operator are defined directly interms of the spin group; one is the standard angular-momentum operator$\mathbf{L}$, while the other is shown to be related to the spin-raisingoperator $\eth$. Computer code is also included, providing an explicitimplementation of the spin-weighted spherical harmonics in this form.

How should spin-weighted spherical functions be defined?