I am wondering how the finite dimensional, irreducible representations of the Lie superalgebra gl(1|1) are parametrized. I understand that they are all highest weight, and that the only non-trivial representations are two dimensional, so they are certainly parametrized by 2-tuples of complex numbers (a,b). The problem I am having is that different literature seems to say different things. Is it required that b-a is a positive integer as in the gl(2) case? This seems to be implied by this paper (2.4 and 2.5).
Or is it the case that $b$ must be real, as seems to be the case in this book (page 157).
I understand this is probably a very simply question, but I'm doubting my own attempts to figure this out considering the literature seems to vary. Any help would be appreciated.