Affine maps between graph isomorphism polytopes and Boolean quadratic polytopes

Let $P_n$ be the set of $n\times n$ permutation matrices. We consider the convex hull $\psi_n$ of the set of tensors $x\otimes x$, $x \in P_n$. We show that $\psi_n$ is a face of the Boolean quadratic polytope $BQP(n^2)$. For every permutation of vertices of the complete graph $K_n$ we consider appropriate $\binom{n}{2} \times \binom{n}{2}$ permutation matrix of the edges of $K_n$. The convex hull $\phi_n$ of all such matrices is an example of the Young polytope $P((n-2,2))$. We show that $\phi_n$ and $\psi_n$ are not isomorphic, but $\phi_n$ is a projection of $\psi_n$. Also, we show that $BQP(n)$ is a face of $\phi_{2n}$.