It's a famous claim that spin-0, spin-1 and spin-2 fields are described by scalar, vector and second-rank tensor, respectively. My question is: why not other objects? For example, consider spin-1 field, we can use a field that carries two left spinor indexes. From the group-theoretic relation we know it contains a scalar component and a spin-1 component, the same as the vector representation. This claim is first recognized (to my knowledge) on a classic paper (Fierz 1939 Helv. Phys. Acta, 12, 3), but it's in German and I can't find the English version online. Can anyone help me understand it? Thanks.