Commutators in finite free probability, I

This paper describes the expected characteristic polynomial of the commutator of randomly rotated matrices, in the context of the finite free probability theory initiated by Marcus, Spielman, and Srivastava. The key technical features are the use of Weingarten calculus to translate the random matrix problem into one of combinatorial representation theory, followed by some applications of the Goulden-Jackson immanant formula and the classical theory of permutation modules.