Noncommutative Birational Rowmotion on Rectangles
The operation of birational rowmotion on a finite poset has been a mainstay in dynamical algebraic combinatorics for the last 8 years. Since 2015, it is known that for a rectangular poset of the form [p]x[q], this operation is periodic with period p+q. (This result, as has been observed by Max Glick, is equivalent to Zamolodchikov’s periodicity conjecture in type AA, proved by Volkov.)
In this talk, I will outline a proof (joint work with Tom Roby) of a noncommutative generalization of this result. The generalization does not quite extend to the full generality one could hope for it covers noncommutative rings, but not semirings; however, the proof is novel and simpler than the original commutative one. Extending this to semirings and to other posets is work in progress.