### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 053, 27 pages      arXiv:1812.08553      https://doi.org/10.3842/SIGMA.2019.053

### Orthogonal Dualities of Markov Processes and Unitary Symmetries

Gioia Carinci a, Chiara Franceschini b, Cristian Giardinà c, Wolter Groenevelt a and Frank Redig a
a) Technische Universiteit Delft, DIAM, P.O. Box 5031, 2600 GA Delft, The Netherlands
b) Center for Mathematical Analysis Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
c) University of Modena and Reggio Emilia, FIM, via G. Campi 213/b, 41125 Modena, Italy

Received December 24, 2018, in final form July 05, 2019; Published online July 12, 2019

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.

Key words: stochastic duality; interacting particle systems; Lie algebras; orthogonal polynomials.

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