Let $G$ be a finite subgroup of the group $U_d(\mathbb{C})$ of unitary transformations of $\mathbb{C}^d$. Suppose that $G$ acts irreducibly but is imprimitive, meaning that there is a nontrivial direct sum decomposition $\mathbb{C}^d = \bigoplus_{i = 1}^r V_i$ such that each $g \in G$ permutes the $V_i$.

Then it seems that the $V_i$ are necessarily orthogonal: I wrote up a proof here. However I'm happy to admit that it took me quite some time to find this proof, and I still don't know of a reference. This must surely be well-known, and was probably known to Frobenius. Can anyone supply me with a reference?