It is the -invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the representation has spin if is half-integral. The simplest are and , the Weyl-spinors of dimension . Then, for example, and are a spin representations of dimensions and respectively. According to the above paragraph, there are subspaces with spin both and in the last two cases, so these representations cannot likely represent a "single" physical particle which must be well-behaved under . It cannot be ruled out in general, however, that representations with multiple subrepresentations with different spin can represent physical particles with well-defined spin. It may be that there is a suitable relativistic wave equation that projects out "unphysical components", leaving only a single spin.