Free Probability and Operators

Date: November 11, 2022

Time: 4:00PM - 5:00PM

Location: BLOC306, ONLINE

Speaker: Jacob Campbell, University of Waterloo

Title: Commutators in finite free probability

Abstract: I will present a recent preprint (arXiv:2209.00523) concerning the commutator in finite free probability. The main result is an explicit formula for the expected characteristic polynomial of AUBU^* - UBU^*A, where A and B are fixed dxd matrices and U is a dxd random unitary matrix, in terms of the respective characteristic polynomials of A and B. The main ideas are to use Weingarten calculus to reduce the problem to one of combinatorial representation theory, and then use a 1992 result of Goulden and Jackson to compute immanants of certain low-rank matrices. Time permitting, I will suggest some potential avenues for finding the d-finite version of the Nica-Speicher formula for the free cumulants of commutators of free variables. ZOOM LINK: https://tamu.zoom.us/j/99940122674