I want to get a better understanding on particle being an irreducible representation. Does that mean one particular type of particles (say particle $A$) is a subspace of the "total" Hilbert space $H$ (which contains all the types?), and the restriction (to the vector space $V_A$ corresponds to particle $A$) of the representation of the Lorentz transformation group (which govern all the transformation of all types of particles) which has no proper subrepresentation.
Or is the space always stay as $H$ no matter what type of the particle we are talking about, then it is just a matter of choosing different representation on $H$, and different representations mean different particles? In another word, I am confused what vector space are we talking about if we say particle $A$ being an irreducible representation. is it the entire $H$? or some subspace $V_A$? Because we say spin $\frac{1}{2}$ corresponds to the two dimensional representation, I am unsure what is two dimensional.