We describe the multiplicative analogue of the celebrated Horn problem where we study the convolution product of two conjugacy classes of the unitary group ???? described by 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.
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