Open Collections

Description

Consider a plane partition $P$ as an order ideal in the product $[a] \times [b] \times [c]$ of three chain posets. The combinatorial rowmotion operator sends $P$ to the plane partition generated by the minimal elements of its complement. What is the orbit structure of this action I will attempt to survey the state of this question. In particular, I will describe my recent work with Becky Patrias, showing that rowmotion exhibits a strong form of resonance with frequency $a+b+c-1$, in the sense that each orbit size shares a prime divisor with $a+b+c-1$. This confirms a 1995 conjecture of Peter Cameron and Dmitri Fon-Der-Flaass.