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

TL;DR

This paper presents a positive, explicit formula for the convolution of conjugacy classes in the unitary group, linking quantum cohomology, polytopes, and flat connection volumes.

Contribution

It introduces a novel positive formula for the product of conjugacy classes in U(n), connecting quantum cohomology and geometric models.

Findings

Provides a subtraction-free sum of polytope volumes for the convolution density.

Yields a positive formula for the volume of SU(n)-valued flat connections.

Connects algebraic convolution with geometric and quantum cohomology structures.

Abstract

The convolution product of two conjugacy classes of the unitary group $U_n$ is described by a probability distribution on the space of central measures. Relating this convolution to the quantum cohomology of Grassmannians and using recent results describing the structure constants of the latter, we give a manifestly positive formula for the density of the probability distribution for the product of generic conjugacy classes. 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. As a consequence, this expression also provides a positive and explicit formula for the volume 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.