I was studying applied group theory to condensed matter, specifically representations.
As far as I undersand, we can represent elements of symmetry (rotations, for example) by matrix, being a matrix representation. If these matrices are irreducible, then the representation is irreducible.
Now, consider an Hamiltonian, $H$, which is invariant under the action of a symmetry element, $R$, of a group. Then,
$$[\hat{P}_R, \hat{H}] = 0$$
and, if $\phi_n$ is an eigenstate of $\hat{H}$ with energy $E_n$ then $\hat{P}_R \phi_n$ is also an eigenstate $\hat{H}$ with the same energy.
My doubt
According to these results, what does it mean for $\phi_n$ to transform as an irreducible representation? I cannot understand the meaning 'transforming states as an irreducible representation'. To my (little) understanding, only symmetry elements like $R$ should transform according to a said representations.