In quantum mechanics, if a system has a symmetry group $G$ representing physical operations we can perform on a system that will leave the evolution unchanged, then this group will act on the states of the system, which are state vectors in the system's Hilbert space $\mathcal{H}$. (The latter, by the way, is unique up to isomorphisms that preserve the operator algebra.)

This group action is equivalent to a homomorphism between $G$ and a group of linear operators on $\mathcal{H}$. (This is not completely trivial in that linearity follows from the physical postulate that the symmetries must also preserve superpositions.) After finding this group representation, then one usually decomposes it into a direct sum of irreducible representations, which give invariant subspaces under the symmetry group, usually labelled by the irrep that acts on them. These are probably the "multiple Hilbert spaces" you refer to.

Essentially, the group representations give a way to act with the symmetries on the actual states of the system. The different ways in which this is possible is studied by representation theory, and it can have a deep impact on the physics of the problem. This, of course, is simply saying that the symmetry of the problem has a deep impact on its solution, which should not be a surprise.

Finally, a group representation that is also an isomorphism is normally called a faithful representation, and it is not necessary for physically meaningful group representations to be faithful or, indeed, even nontrivial. If you look at the representations of the rotation group, $SO(3)$, you find at its lowest dimension the invariant subspace $j=0$ with the unit representation: completely trivial action, which does not stop us using it!

This post imported from StackExchange Physics at 2015-03-30 13:55 (UTC), posted by SE-user Emilio Pisanty