Birational and noncommutative lifts of antichain toggling and rowmotion
Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 955-984.

The rowmotion action on order ideals or on antichains of a finite partially ordered 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 of these nice features still hold when the action is extended to [0,1]-labelings of the poset or (via detropicalization) to labelings by rational functions (the birational setting).

In this work, we parallel the birational lifting already done for order-ideal rowmotion to antichain rowmotion. We give explicit equivariant bijections between the birational toggle groups and between their respective liftings. We further extend all of these notions to labellings by noncommutative rational functions, setting an unpublished periodicity conjecture of Grinberg in a broader context.

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DOI: 10.5802/alco.125
Classification: 05E18, 06A07, 12E15
Keywords: Antichain, birational rowmotion, dynamical algebraic combinatorics, graded poset, homomesy, isomorphism, noncommutative algebra, periodicity, rowmotion, toggle group, transfer map.

Joseph, Michael 1; Roby, Tom 2

1 Department of Technology and Mathematics Dalton State College 650 College Dr. Dalton, GA 30720, USA
2 University of Connecticut 341 Mansfield Road Storrs, CT 06269-1009, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Joseph, Michael; Roby, Tom. Birational and noncommutative lifts of antichain toggling and rowmotion. Algebraic Combinatorics, Volume 3 (2020) no. 4, pp. 955-984. doi : 10.5802/alco.125. https://alco.centre-mersenne.org/articles/10.5802/alco.125/

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