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

Nizar Demni; Tarek Hamdi

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arXiv · 1906.07949

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

Nizar Demni·Tarek Hamdi

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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.

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BibTeX RIS

Nizar Demni, Tarek Hamdi. 2019-06-19. Schur-Weyl duality and the Product of randomly-rotated symmetries by a unitary Brownian motion. https://arxiv.org/abs/1906.07949

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