It is the SO(3) -invariant subspaces of the irreducible representations that determine whether a representation has spin. From the above paragraph, it is seen that the (m, n) representation has spin if m + n is half-integer. The simplest are (⁠ 1 / 2 ⁠, 0) and (0, ⁠ 1 / 2 ⁠), the Weyl-spinors of dimension 2. Then, for example, (0, ⁠ 3 / 2 ⁠) and (1, ⁠ 1 / 2 ⁠) are a spin representations of dimensions 2⋅ ⁠ 3 / 2 ⁠ + 1 = 4 and (2 + 1)(2⋅ ⁠ 1 / 2 ⁠ + 1) = 6 respectively. According to the above paragraph, there are subspaces with spin both ⁠ 3 / 2 ⁠ and ⁠ 1 / 2 ⁠ in the last two cases, so these representations cannot likely represent a single physical particle which must be well-behaved under SO(3). It cannot be ruled out in general, however, that representations with multiple SO(3) 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.