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Explore the joint behavior of independent random tensor matrices in this lecture from the Workshop on "Non-commutative Geometry meets Topological Recursion" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into the extension of free probability theory to cases with smaller symmetry groups, focusing on matrices with tensor structure. Examine two key topics: the case where invariants are tensors of local unitaries, including generalizations of unitary Weingarten calculus and HCIZ integrals, and the case where symmetries are an image of the unitary group under an irreducible representation. Discover new techniques for studying analytic properties of matrices, such as typical operator norm, and explore open questions in the field of free probability theory.
