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.

• Have you illustrated all these statements with the rotation group you mastered in college? Jan 13, 2021 at 16:09
• @CosmasZachos This was explained to me in a rather abstract way. Jan 13, 2021 at 16:12
• A representation of $G$ is a map $D(g)$ which for every $g\in G$ associates one linear operator in some vector space $V$ and which reproduces the group composition law $D(gh)=D(g)D(h)$. Saying that an object transform according to some representation means that the object is an element of the vector space on which the group acts according to the representation.
– Gold
Jan 13, 2021 at 16:13
• I think I got it, thanks @user1620696! Jan 13, 2021 at 16:17

When we say that an eigenstate $$\phi_n$$ transforms as an irrep $$\rho$$ of a group $$G$$, we mean that it belongs to a subspace of the full Hilbert space which is mapped onto itself under the action of $$\rho$$ (in the present context, this subspace is an eigenspace for a particular Hamiltonian eigenvalue). That is, $$\phi_n$$ belongs to a subspace $$V$$ such that for any $$\rho(g)$$ for $$g\in G$$, $$\rho(g)\phi_n \in V$$