A major achievement of free probability has been to describe the joint behavior of independent random matrices whose distributions are invariant under unitary conjugation. It is natural to wonder how this can be extended when the symmetry group is smaller. We focus on the case where the matrices have a tensor structure and the symmetries are given by tensors of groups chosen accordingly.
These lectures will describe the following topics:
1- the case where the invariants are tensors of local unitaries (based on a series of joint works with Gurau and Lionni). This includes the study of generalizations of the unitary Weingarten calculus and HCIZ integrals.
2- the case where the symmetries are an image of the unitary group under an irreducible representation (based on joint work with Bordenave).
The first part proposes generalizations of free probability theory and comes with multiple open questions. The second parts sticks within the realm of free probability theory but proposes new techniques to study analytic properties of matrices, such as their typical operator norm.