In Spin representations post and
in Rotations and Lorentz groups post is presented how the representations of the restricted Lorentz group are defined by the generators corresponding to rotations and boosts generators of the restricted Lorentz group. The fact that any representation of the restricted Lorentz group is a direct sum of irreducible representations is a consequence of a more general result which states that any representation of a semisimple Lie algebra is a direct sum of irreducible representations. However, in the “Lorentz group representations pdf” file embedded below is presented a proof of this fact that reveals the structure of general representations of the restricted Lorentz group, allowing the clasification of irreducible representations of the restricted Lorentz group as double indexed spin (j , k) representations from which those with integer j + k are characteristic for bosons and those with half integer j + k are characteristic for fermions as shown further in the “Spin-statistics pdf” file embedded below , that also presents a proof of the Spin-statistics theorem, revealing the fact that, depending on the nature of the characterizing representation, particles systems obey the Bose-Einstein statistics or Fermi-Dirac statistics, which can be seen as a consequence of Lorentz invariance.
Lorentz group representations pdf
These two chapters you can also find in the post
Notes on … quantum field theory