# Orthogonal Dualities of Markov Processes and Unitary Symmetries

## TL;DR

This paper explores the connection between orthogonal self-dualities in interacting particle systems and unitary symmetries of their Markov generators, introducing new methods to derive duality functions.

## Contribution

It reveals that orthogonal dualities originate from unitary symmetries and introduces a novel approach using scalar products to derive duality functions.

## Key findings

- Orthogonal dualities are linked to unitary symmetries of the Markov generator.
- Two equivalent expressions for symmetries are provided, related by the Baker-Campbell-Hausdorff formula.
- A new method using scalar products is introduced to derive duality functions.

## Abstract

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.