Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass

TL;DR

This paper studies a special type of combinatorial dynamics called NRP rowmotion on posets, proving it for minuscule posets and showing its dependence only on the comparability graph, extending previous results on product of chains.

Contribution

It establishes NRP rowmotion for all minuscule posets and shows that NRP promotion depends solely on the isomorphism class of the comparability graph.

Findings

NRP rowmotion holds for all minuscule posets

NRP promotion depends only on the comparability graph

Extends previous results from product of chains to minuscule posets

Abstract

Let $P$ be a graded poset of rank $r$ and let $\mathbf{c}$ be a $c$-element chain. For an order ideal $I$ of $P \times \mathbf{c}$, its rowmotion $\psi(I)$ is the smallest ideal containing the minimal elements of the complementary filter of $I$. The map $\psi$ defines invertible dynamics on the set of ideals. We say that $P$ has NRP ("not relatively prime") rowmotion if no $\psi$-orbit has cardinality relatively prime to $r+c+1$. In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and D. Fon-Der-Flaass by establishing NRP rowmotion for the product $P = \mathbf{a} \times \mathbf{b}$ of two chains, the poset whose order ideals correspond to the Schubert varieties of a Grassmann variety $\mathrm{Gr}_a(\mathbb{C}^{a+b})$ under containment. Here, we initiate the general study of posets with NRP rowmotion. Our first main result establishes NRP rowmotion for all minuscule posets $P$, posets whose order ideals reflect the Schubert stratification of minuscule flag varieties. Our second main result is that NRP promotion depends only on the isomorphism class of the comparability graph of $P$.