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The rowmotion action on order ideals or on antichains of a finite partiallyordered set has been studied (under a variety of names) by many authors.Depending on the poset, one finds unexpectedly interesting orbit structures,instances of (small order) periodicity, cyclic sieving, and homomesy. Many ofthese nice features still hold when the action is extended to $[0,1]$-labelingsof the poset or (via detropicalization) to labelings by rational functions (thebirational setting).  In this work, we parallel the birational lifting already done for order-idealrowmotion to antichain rowmotion. We give explicit equivariant bijectionsbetween the birational toggle groups and between their respective liftings. Wefurther extend all of these notions to labellings by noncommutative rationalfunctions, setting an unpublished periodicity conjecture of Grinberg in abroader context.

Birational and noncommutative lifts of antichain toggling and rowmotion