Positive formula for the product of conjugacy classes on the unitary group

Quentin François (1, 2) , Pierre Tarrago (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 PSL - Université Paris Sciences et Lettres
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
4 SU - Sorbonne Université

The convolution product of two generic conjugacy classes of the unitary group $U_n$ is 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 manifestly positive formula for this density. In the same vein as the hive model of Knutson and Tao, this formula is given in terms of a subtraction-free sum of volumes of explicit polytopes. As a consequence, this expression also provides a positive formula for the volume of moduli spaces of $SU_n$-valued flat connections on the three-holed two dimensional sphere, which was first given by Witten in terms of an infinite sum of characters.

Mots clés: Convolution of conjugacy classes; Unitary group; Quantum cohomology of Grassmannians; Volume of moduli spaces

Domaines: Probabilités [math.PR]; Combinatoire [math.CO]; Physique mathématique [math-ph]; Géométrie symplectique [math.SG]

Quentin François, Pierre Tarrago. Positive formula for the product of conjugacy classes on the unitary group. 2024. ⟨hal-04572895⟩