I was trying to make a 'dictionary' between group action and group representation terms using the $\mathbb{C}[-]$ functor. I immediately found that if the set $Y \subset X$ is invariant under the action of $G$, then the corresponding representation is reducible. This of course means that irreducible represenations cannot be born out of actions that have non-trivial invariant subsets. But then again, the action of $\mathbb{Z}/2^2$ on itself by left shifts is transitive, but the corresponding representation is of course reducible. More generally, if $G$ is abelian then it appears that no irreducible representation can come out of group action.
My question is: does it make sense to try and draw parallels further? I find it compelling that the functor takes $\sqcup$ to $\oplus$ and $\times$ to $\otimes$ (and these operations turn out to be important in representation theory), but the lack of an analog for irreducibility means that most representation theory will not correspond to anything.