Birational rowmotion on a rectangle over a noncommutative ring

Abstract: We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained. Consider labellings of a finite poset $P$ by $\left|P\right| + 2$ elements of a ring $\mathbb{K}$: one label associated with each poset element and two constant labels for the added top and bottom elements in $\hat{P}$. Birational rowmotion is a partial map on such labellings. It was originally defined by Einstein and Propp for $\mathbb{K}=\mathbb{R}$ as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope $\mathcal{O}(P) := {\text{order-preserving } f: P \to[0,1]}$. The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of $P$, which correspond to the vertices of $\mathcal{O}(P)$. Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing $\mathbb{K}$ to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step. In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when $P$ is a product of two chains) for $\mathbb{K}$ a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.