On projection operators for irreducible representations of finite groups Physicist here, posting here because it seems more appropriate.
I will try to pose this question in the most general way possible, so that it is applicable to the most broad number of situations. If it is at all relevant for the answer keep in mind that this question arises from the study of the rotational symmetry of quantum operators on a lattice.
I have a system with a symmetry described by a finite group $G$ of order $n_G$. (the point group in question is the cubic group $O$ of order 24 (it is isomorphic to $S_4$)).
I have a set of objects (operators) $\{\hat{O}_i\}$ spanning a vector space of operators, we will call $H = \text{Span}\{\hat{O}_i\}$.
I have already managed by hand to build the reducible representation $\mathcal{R}$ acting on the $\{\hat{O}_i\}$ basis. I already know ho to compute the Kronecker decomposition of $\mathcal{R}$ by calculating the multiplicities of each irreducible representation of the group using the character tables.
I need to find linear combinations of these $\{\hat{O}_i\}$:
$$
\overline{O}^\mu_a = \sum_i c^\mu_{a,i}\hat{O}_i
$$
Such that $\overline{O}^\mu_a$ transforms according to an irreducible representation of the point group symmetry. In particular, if $\Gamma^\mu$ is the $\mu$-th irreducible representation of the finite group then for every fixed $\mu$ the set $\{\overline{O}^\mu_a \text{ with } a= 1, ..., \dim \Gamma^\mu\}$ is an orthonormal basis of the invariant subspace of $H$ that tranforms according to the $\Gamma^\mu$ representation.
In literature I have found that we can do this using projection operators and both the objects:
$$
\frac{\dim \Gamma^\mu}{n_G}\sum_{g\in G} {\Gamma^\mu}(g)^\dagger_{ab} \mathcal{R}(g)_{ij}\hat{O}_j \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \text{and} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \sum_{g \in G} \chi^\mu(g)^*\mathcal{R}(g)_{ij}\hat{O}_i
$$
should transform according to the $\Gamma^\mu$ representation because of the great orthogonality theorem. The difference in the two approaches is essentially that the first projector is also able to extract an orthonormal basis of the $\Gamma^\mu$ subspace of $H$.

My question is the following, if the Kronecker decomposition of $H$ has different copies of any irreducible representation, i.e. if we have that:
$$
\mathcal{R} = \bigoplus_\mu a_\mu \Gamma^\mu
$$
where there exist $\nu$ such that $a_\nu \geq 2$. Then on which of the subspaces am I projecting? If there are multiple copies of the $\Gamma^\nu$ irreducible representation, and I wish to build an orthonormal basis for each separate invariant subspace tranforming according to $\Gamma^\nu$ then how should I interpret the action of the projection operators? (The first one in particular since I am looking to build the linear combinations explicitly so that would seem more appropriate).
I hope the question is undestandable, but I remain available to provide any clarification if necessary.
Thanks to all in advance.
 A: As I am not trained in physics, I don't quite understand your notation, but I think I understand enough to guess what you're asking for. I am only able to answer in a notation typical in mathematics, though.
Let $(V,\rho)$ be a representation of a finite group. Then we have a decomposition $$V=\bigoplus_i V_i^{n_i}$$ where $V_i$ runs over all (pairwise non-isomorphic) irreducible representations. (I guess this is what you call the Kronecker decomposition)
Now if $V_i$ is an irreducible representation with character $\chi_i$, then we have the isotypical projection operator
$$P_i:v \mapsto \frac{\mathrm{dim}V_i}{|G|}\sum_{g \in G}\overline{\chi(g)}\rho(g)v$$
What this operator does is that it is a $G$-equivariant projection from $V$ onto $V_i^{n_i}$, the latter is called the $V_i$-isotypical component of $V$.
If $n_i \geq 2$, then the image of this projection operator is not irreducible, but it is a direct sum of copies of the same irreducible representation.
Depending on what you mean by the phrase "to transform according to the irreducible representation $V_i$", it may or may not be true that the image of the projection operator does this:

*

*If you mean that the image of $P_i$ is isomorphic to $V_i$ as a representation, then this is generally not true.

*If you mean that the image of $P_i$ can be decomposed into representations isomorphic to $V_i$, then it is true.

In the latter case, the image of $P_i$ is the maximal subspace that "transforms according to $V_i$".
