It makes sense that Polchinski is putting Weyl spinors in B.1.44 and B.6.3. As fzero pointed out by checking the chirality. However, I was told the number in formula like B.1.44 refers to the dimensions of real irreducible representation for the corresponding algebra/subalgebra. Since so(5,1)...
The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub-symmetry. I'm learning it from the appendix B in Polchinski's big book...