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 We further extend all of these notions to labellings by noncommutative rational functions, setting an unpublished periodicity conjecture of Grinberg in a broader context.

Revised:

Accepted:

Published online:

Keywords: Antichain, birational rowmotion, dynamical algebraic combinatorics, graded poset, homomesy, isomorphism, noncommutative algebra, periodicity, rowmotion, toggle group, transfer map.

^{1}; Roby, Tom

^{2}

@article{ALCO_2020__3_4_955_0, author = {Joseph, Michael and Roby, Tom}, title = {Birational and noncommutative lifts of antichain toggling and rowmotion}, journal = {Algebraic Combinatorics}, pages = {955--984}, publisher = {MathOA foundation}, volume = {3}, number = {4}, year = {2020}, doi = {10.5802/alco.125}, language = {en}, url = {http://www.numdam.org/articles/10.5802/alco.125/} }

TY - JOUR AU - Joseph, Michael AU - Roby, Tom TI - Birational and noncommutative lifts of antichain toggling and rowmotion JO - Algebraic Combinatorics PY - 2020 SP - 955 EP - 984 VL - 3 IS - 4 PB - MathOA foundation UR - http://www.numdam.org/articles/10.5802/alco.125/ DO - 10.5802/alco.125 LA - en ID - ALCO_2020__3_4_955_0 ER -

%0 Journal Article %A Joseph, Michael %A Roby, Tom %T Birational and noncommutative lifts of antichain toggling and rowmotion %J Algebraic Combinatorics %D 2020 %P 955-984 %V 3 %N 4 %I MathOA foundation %U http://www.numdam.org/articles/10.5802/alco.125/ %R 10.5802/alco.125 %G en %F ALCO_2020__3_4_955_0

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. http://www.numdam.org/articles/10.5802/alco.125/

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