Free probability of type B prime

Free probability of type $B$ was invented by Biane-Goodman-Nica, and then it was generalized by Belinschi-Shlyakhtenko and Février-Nica to infinitesimal free probability. The latter found its applications to eigenvalues of perturbed random matrices in the work of Shlyakhtenko and Cébron-Dahlqvist-Gabriel. This paper offers a new framework, called ``free probability of type $B$ prime'', which appears in the large size limit of independent unitarily invariant random matrices with perturbations. Our framework is related to boolean, free, (anti)monotone, cyclic-(anti)monotone, and conditionally free independences. We then apply the new framework to the principal minor of unitarily invariant random matrices, which leads to the definition of a multivariate inverse Markov-Krein transform. This talk is based on a joint work with Katsunori Fujie.