MIT-Harvard-MSR Combinatorics Seminar

Speaker:  Darij Grinberg (Drexel)

Title:  Noncommutative Birational rowmotion on Rectangles

Abstract:

Birational rowmotion is a (partial) map defined for any field and any finite poset, generalizing some combinatorial and geometric constructions.

Since 2015, it is known that for a rectangular poset of the form $[p] \times [q]$, this map 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 gives rise to some interesting problems (work in progress).