Darij Grinberg: Noncommutative Birational Rowmotion on Rectangles

Abstract: Birational rowmotion on a finite poset has been a mainstay in dynamical algebraic combinatorics for a decade. Since 2015, it is known that for a rectangular poset of the form $[p] \times [q]$, 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; it also generalizes a known property of Young tableau promotion due to Schützenberger.)

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.

Some open questions will be discussed, including a possible generalization to semirings and variants for other posets.