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 $\mathsf{B}$. Let $\Psi (P)$ denote the smallest plane partition containing the minimal elements of $\mathsf{B}-P$. 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.

6 pages. Title updated again to match publication version