Dynamics of plane partitions: Proof of the Cameron-Fon-Der-Flaass conjecture

Abstract

One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995). Consider a plane partition $P$ in an $a \times b \times c$ box ${\sf B}$. Let $\Psi(P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$. Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the $\Psi$-orbit of $P$ is always a multiple of $p$. This conjecture was established for $p \gg 0$ by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of $p$ in work of K. Dilks, J. Striker, and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.