Strong convergence for sequences of representations of SU(2) with growing dimension

Establish strong convergence to free Haar unitaries for random matrices U_k^N = π_N(g_k), where g_k are i.i.d. Haar-distributed in SU(2) and π_N:SU(2)→U(D_N) are irreducible unitary representations with D_N→∞, i.e., prove that lim_{N→∞} ||P(U_1^N, …, U_r^N)|| = ||P(u_1, …, u_r)|| for every *-polynomial P and free Haar unitaries u_1, …, u_r.

Background

Existing strong convergence results typically involve growing groups (e.g., S_N) or ensembles with substantial randomness. The authors ask whether strong convergence can be obtained when the group is fixed, such as SU(2), and only the representation dimension diverges.

Resolving this would extend strong convergence to settings motivated by representation theory of compact Lie groups and could impact geometric and operator-algebraic applications.