Schur's lemma and average over pure states

Question

How does Schur's lemma prove Claim 9 of

this article? The claim is that for pure states $\eta$ on some Hilbert space $\mathcal{H}$, $$\int_{\eta}\eta^{\otimes n} d\eta$$ is a multiple of identity on the symmetric subspace (permutation invariant subspace) of $\mathcal{H^{\otimes n}}$. Because the integral is a quantum state this multiple is the maximally mixed state on $Sym^{(n)}(\mathcal{H})$. Schur's lemma as far as I know states that if a non-zero matrix commutes with all the matrices in an irreducible representation of a group, then it is a multiple of identity. \

This claim (claim 9 of the above mentioned article) seems to be well known as there are variations of it all over the literature. I appreciate your input.

Answer 1

Let $M = \int_{\eta}\eta^{\otimes n} d\eta$. The operators $\{P_d(\pi) \mid \pi \in S_n\}$ form an irreducible representation of $S_n$ when restricted to their common invariant subspace $Sym^{(n)}(\mathcal H)$.

Thus, by Schur's lemma, all we have to do is show that $M$ commutes with $P_d(\pi)$ for all $\pi \in S_n$. This just amounts to noting that linear operators commute with integration: $$ P_d(\pi) M = P_d(\pi) \int_{\eta}\eta^{\otimes n} d\eta = \int_{\eta}P_d(\pi)\eta^{\otimes n} d\eta \overset!{=} \int_{\eta}\eta^{\otimes n} P_d(\pi)d\eta = MP_d(\pi) $$