Strong asymptotic freeness for independent uniform variables on compact groups associated to nontrivial representations

Abstract

Voiculescu discovered asymptotic freeness of independent Haar-distributed unitary matrices. 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 nontrivial 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 with 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. 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.