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The discrete periodic Pitman transform: invariances, braid relations, and Burke properties
Eva R. Engel, Benjamin Jasper Kra-Caskey, Oleksandr Lazorenko, Caio Hermano Maia de Oliveira, Evan Sorensen, Ivan Wong, Ryan Xu, Xinyi Zhang
https://arxiv.org/abs/2508.05603

The discrete periodic Pitman transform: invariances, braid relations, and Burke properties
We develop the theory of the discrete periodic Pitman transform, first introduced by Corwin, Gu, and the fifth author. We prove that, for polymers in a periodic environment, single-path and multi-path partition functions are preserved under the action of this transform on the weights in the polymer model. As a corollary, we prove that the discrete periodic Pitman transform satisfies the same braid relations that are satisfied for the full-line Pitman transform, shown by Biane, Bougerol, and O'C…