What is meant by a vector representation?

Question

Physicists often talk about a vector representation. For example, the first comment to this answer says:

That the $(1/2,1/2)$ representation corresponds to a vector irreducible representation of the Lorentz group is not obvious.

From a mathematical point of view all representation are valued in vector spaces so it's somewhat confusing to talk of vector representations. That is all reps are

$G \rightarrow Aut V$

Where $V$ is some vector space.

My first inclination is that it's simply a physicists way of speaking to distinguish between spinorial reps (which, rather than going from G, goes from the universal cover of G, which in the case of the Lorentz group is its double cover) and ordinary 'vector' representations, which goes directly from G.

Is this on the right track, or is there more to calling a rep, a vector rep?

Answer 1

A similar thing happens in mathematics as well. Although $V\otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^*$ is a vector space, in many contexts its elements are not called vectors but tensors, whereas the name vector is reserved for elements of $V$.

For the Lorentz group we have, for example, the trivial representation, which we call a scalar (even if it is a vector, trivially). The spinor representations are best identified as spinors rather than as vectors (although they are vectors, of course). We can use the name vector for the representation $(1/2,1/2)$, because it acts on the vectors in the physical space-time. For higher spin, we have other specific names to identify the representations. For example, just as in mathematics, the tensor products of our vector representation are usually called tensors.