Sylvain Lacroix: Finite N precursors of the free cumulants

mercredi 14 janv. 2026, 14:00 → 15:00 Europe/Paris

In this talk, I will discuss various properties of a family of finite N precursors of the free cumulants. They are polynomials on the space of NxN matrices, invariant under conjugations and extracted from the so-called HCIZ integral. Their main feature is their additivity with respect to the average over sums of U(N)-conjugacy orbits. Moreover, they admit natural expansions in terms of Newton and Schur polynomials, related to the representation theory of the unitary and symmetric groups and recovering a previous construction by Capitaine and Casalis. I will further discuss their large N limit, which yields the free cumulants, with 1/N² corrections expressed in terms of Hurwitz numbers. Finally, expanding on the main additivity property of these objects, I will describe the behaviour of general U(N)-invariant polynomials under sums of conjugacy orbits. This talk is based on arXiv:2508.21483, in collaboration with Jean-Bernard Zuber.