Probabilistic Operator Algebra Seminar: Semiclassical asymptotics of $GL_N(\mathbb C)$ tensor products

Seminar | March 20 | 3-5 p.m. | 736 Evans Hall

 Jonathan Novak, UC San Diego

 Department of Mathematics

The existence of a link between the asymptotics of tensor products of irreducible representations of $GL_N(\mathbb C)$ and free probability was discovered by Biane in the 90s. Biane realized that there is a matrix model for the "Littlewood-Richardson process" encoding the invariant subspaces which occur in the tensor product, and that the (quantum) random matrices involved in this model become asymptotically free in the large-dimension limit. Biane's arguments depend on a semiclassical parameter which degenerates quantum random matrices to classical random matrices in the large-dimension limit. His arguments required superlinear decay of this parameter. Recently, Bufetov and Gorin conjectured that linear decay is sufficient. In recent joint work with Collins and Sniady, we proved that *any* decay rate of the semiclassical parameter suffices to see the emergence of free probability in the asymptotics of $GL_N(\mathbb C)$ tensor products. Thus the connection between asymptotic representation theory and free independence is much more robust than previously thought.