Resonance in orbits of plane partitions and increasing tableaux

摘要

We introduce a new concept of resonance on discrete dynamical systems. Thisconcept formalizes the observation that, in various combinatorially-naturalcyclic group actions, orbit cardinalities are all multiples of divisors of afundamental frequency. Our main result is an equivariant bijection between plane partitions in a box(or order ideals in the product of three chains) under rowmotion and increasingtableaux under $K$-promotion. Both of these actions were observed to have orbitsizes that were small multiples of divisors of an expected orbit size, and weshow this is an instance of resonance, as $K$-promotion cyclically rotates theset of labels appearing in the increasing tableaux. We extract a number ofcorollaries from this equivariant bijection, including a strengthening of atheorem of [P. Cameron--D. Fon-der-Flaass '95] and several new results on theorder of $K$-promotion. Along the way, we adapt the proof of the conjugacy ofpromotion and rowmotion from [J. Striker--N. Williams '12] to give ageneralization in the setting of $n$-dimensional lattice projections. Finallywe discuss known and conjectured examples of resonance relating to alternatingsign matrices and fully-packed loop configurations.