Minimal randomness for strong convergence in irreducible representations of the symmetric group

Determine whether strong convergence to free Haar unitaries holds for the random matrices U_k^N = π_N(σ_k^N), where σ_1^N, …, σ_r^N are i.i.d. uniformly distributed in the symmetric group S_N and π_N:S_N→U(D_N) is any irreducible unitary representation with D_N>1, in the sense 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

The paper establishes strong convergence for a wide class of random matrix models derived from group representations and highlights Cassidy’s theorem, which proves strong convergence for many irreducible representations of S_N up to dimension exp(N^{1/21}). This dramatically reduces randomness relative to classical models.

The authors ask how far such reductions can go: specifically, whether strong convergence might hold for any irreducible representation sequence with D_N>1, regardless of growth constraints. Resolving this would clarify the minimal randomness needed to achieve strong convergence in permutation-based models.