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Asymptotic freeness of independent Haar distributed unitary matrices wasdiscovered by Voiculescu. Many refinements have been obtained, including strongasymptotic freeness of random unitaries and strong asymptotic freeness ofrandom permutations acting on the orthogonal of the Perron-Frobeniuseigenvector. In this paper, we consider a new matrix unitary model appearingnaturally from representation theory of compact groups. We fix a non-trivialsignature $\rho$, 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 inthis generalized context when drawing independent copies of the Haar measure.We also obtain the orthogonal variant of this result. Thanks to classicalresults in representation theory, this result is closely related to strongasymptotic freeness for tensors, which we establish as a preliminary. In orderto achieve this result, we need to develop four new tools, each of independenttheoretical interest: (i) a centered Weingarten calculus and uniform estimatesthereof, (ii) a systematic and uniform comparison of Gaussian moments andunitary moments of matrices, (iii) a generalized and simplified operator valuednon-backtracking theory in a general $C^*$-algebra, and finally, (iv)combinatorics of tensor moment matrices.

Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations