Séminaire Lotharingien de Combinatoire, 89B.50 (2023), 12 pp.

Darij Grinberg and Tom Roby

Birational Rowmotion over Noncommutative Rings

Abstract. We study the dynamics of birational rowmotion over an arbitrary noncommutative ring K. This generalizes the birational rowmotion map in the commutative setting, which itself lifts the well-studied combinatorial rowmotion map on a finite poset. When the underlying poset P is a rectangle (i.e., a product of two chains), this operation has "twisted periodicity" and "reciprocity" properties, surprisingly similar to the commutative setting. We briefly outline proofs of these results (details are on the arXiv) and discuss extensions and variants. In particular, we conjecture similar results for the case when P is a Δ- or ∇-shaped triangle or a trapezoid. We also conjecture that the results remain valid when K is a semiring. We further prove some elementary properties of birational rowmotion for general P, and (for the sake of exposition) discuss connections to the octahedron recurrence and Zamolodchikov periodicity (which are not new, but deserve better circulation).

Received: November 15, 2022. Accepted: February 20, 2023. Final version: April 1, 2023.

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