  Asymptotic freeness of independent Haar distributed unitary matrices was
discovered by Voiculescu. Many refinements have been obtained, including strong
asymptotic freeness of random unitaries and strong asymptotic freeness of
random permutations acting on the orthogonal of the Perron-Frobenius
eigenvector. In this paper, we consider a new matrix unitary model appearing
naturally from representation theory of compact groups. We fix a non-trivial
signature $\rho$, i.e. two finite sequences of non-increasing natural numbers,
and for $n$ large enough, consider the irreducible representation $V_{n,\rho}$
of $\mathbb{U}_n$ associated to the signature $\rho$. We consider the quotient
$\mathbb{U}_{n,\rho}$ of $\mathbb{U}_n$ viewed as a matrix subgroup of
$\mathbb{U}(V_{n,\rho})$, and show that strong asymptotic freeness holds in
this generalized context when drawing independent copies of the Haar measure.
We also obtain the orthogonal variant of this result. Thanks to classical
results in representation theory, this result is closely related to strong
asymptotic freeness for tensors, which we establish as a preliminary. In order
to achieve this result, we need to develop four new tools, each of independent
theoretical interest: (i) a centered Weingarten calculus and uniform estimates
thereof, (ii) a systematic and uniform comparison of Gaussian moments and
unitary moments of matrices, (iii) a generalized and simplified operator valued
non-backtracking theory in a general $C^*$-algebra, and finally, (iv)
combinatorics of tensor moment matrices.
