How do states in Hilbert Space act like irreducible representations?

Question

I am reading Georgi's book on group theory and I came across this sentence..." Hilbert space of any parity invariant system can be decomposed into states that behave like irreducible representations". I do not understand what does he mean by "states behaving like irreducible representation". Is he just referring to the fact that the dimensionality of the invariant subspace is lower than the full vector space V, just like dimensionality of an irreducible representation is lower than that of the full reducible representation of the group ? Or is there is anything deeper that is going on here ?

I understand, for example, that the parity group's infinite-dimensional representation (acting on Hilbert Space) is reducible to a one-dimensional irreducible representation. So my thinking here would be that the Hilbert space could be decomposed into a one-dimensional invariant subspace....which is enough (and convenient) to study the properties of this Hilbert space under the action of parity symmetry group.

I have read this post on stackexchange Hilbert space decomposition into irreps and understand all the mathematics there. But it was not clear to me from this post how states behave like an irreducible representation of the group ?

Thank you for your time

Answer 1

Whenever you have a symmetry group $G$, it means that for each $g\in G$ there is an operator $U(g)$ (usually unitary) in the system corresponding to the action of $g$. "states behave like irrep of $G$" means that the state space can be organized into subspaces, and in each subspace $U(g)$ form an irrep of G (i.e. the matrix representation of $U(g)$).