In quantum information, much can be done with the averaging formula $$ \int_{\mathbb{C}P^{n-1}} (zz^*)^{\otimes t} dz = {n + t -1 \choose t}^{-1} \operatorname{\Pi}_{\mathrm{Sym}^t}$$ Here the integral gives an average over all unit vectors $z \in \mathbb{C}^n$ and $\operatorname{\Pi}_{\mathrm{Sym}^t}$ denotes the orthogonal projector onto the totally symmetric subspace of $(\mathbb{C}^n)^{\otimes t}$. This result appears as Lemma 1 in [the following paper][1]. Does this result have an analog for the average over all unit vectors $x \in \mathbb{R}^n$? $$ \int_{\mathbb{R}P^{n-1}} (xx^\top)^{\otimes t} dx = {?} $$ By direct computations, I see that the first formula does not hold when $\mathbb{C}P$ is replaced by $\mathbb{R}P$ for $t=2$. Having a formula for $\int_{\mathbb{R}P^{n-1}} (xx^\top)^{\otimes t} dx$ (even for $t = 2$) would help with a problem I'm working on, but I have not been able to find a reference for this or discover a formula myself. [1]: https://arxiv.org/pdf/quant-ph/0604049.pdf