I have the following information: $T$ is the one-dimensional quaternion vector space with the canonical action of $\Gamma$, a finite subgroup of SU$(2)$. This makes sense as SU$(2)$ is the unit quaternions, so its action on $T$ is just multiplication by a quaternionic scalar. So I understand that. Now I have some equation with $S^+$ saying $S^+$ is the positive half-spin space of the vector space $T$. I'm confused on this point because I thought you have a choice with spin structures. So how can there be a "the" positive half-spin space? Perhaps this is a misuse of "the" on behalf of the authors and/or perhaps they are implicitly assuming we've chose our spin structures?
I'll also openly admit that spin structures confuse the heck out of me so maybe I'm totally missing something. Any help would be greatly appreciated.
There can different spin structures on a vector bundle. But for a vector space (i.e. a bindle over one point) the only thing to “choose” for a spin structure is orientation. In 4 dimensions it is equivalent to a $W = W^+ \oplus W^-$ decomposition to eigenspaces (wrt representation of the geometric algebra) of the chiral element ω – then, the choice of sign for ω determines what is $W^+$ and what is $W^-$.
A complex or quaternionic vector space always has canonical (real) orientation. Note that to build half-spin spaces you have to complexify it, and then split 4 complex dimensions in halves. Can’t produce explicit formulae since insufficient information is given (namely, wasn’t specified whether quaternionic module T is left or right).