Quentin François: A positive formula for the product of conjugacy classes on the unitary group

Abstract: We describe from a probabilistic viewpoint the convolution product of two conjugacy classes of the unitary group U(n). The description is given in terms of a probability distribution on the space of central measures which admits a density. Relating the convolution to the quantum Littlewood-Richardson coefficients and using recent results describing those coefficients, we give a positive formula for this density. In the same flavor as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. We will also connect the probability density to the computation of the volume of the moduli space of flat SU(n) connections and to the Yang-Mills partition function. This is based on a joint work with Pierre Tarrago (ArXiv:2405.06723).