# Schur-Weyl duality and the Product of randomly-rotated symmetries by a unitary Brownian motion

## TL;DR

This paper introduces a novel unitary matrix process related to Hermitian matrix-Jacobi processes, deriving its moments through stochastic calculus and symmetric group actions, and establishing a connection with independent unitary Brownian motions.

## Contribution

It presents a new process combining deterministic symmetries and random rotations, with explicit moment formulas and a novel proof technique involving unitary bridges.

## Key findings

- Derived an autonomous ODE for moments of the process.
- Expressed moments via a unitary bridge with an independent Brownian motion.
- Provided a second proof of the moment formulas using the bridge.

## Abstract

In this paper, we introduce and study a unitary matrix-valued process which is closely related to the Hermitian matrix-Jacobi process. It is precisely defined as the product of a deterministic self-adjoint symmetry and a randomly-rotated one by a unitary Brownian motion. Using stochastic calculus and the action of the symmetric group on tensor powers, we derive an autonomous ordinary differential equation for the moments of its fixed-time marginals. Next, we derive an expression of these moments which involves a unitary bridge between our unitary process and another independent unitary Brownian motion. This bridge motivates and allows to write a second direct proof of the obtained moment expression.