Resonance in orbits of plane partitions and increasing tableaux

TL;DR

This paper introduces the concept of resonance in cyclic group actions on combinatorial objects, establishing a bijection between plane partitions and increasing tableaux, and explores implications for orbit sizes and related combinatorial structures.

Contribution

It formalizes resonance in combinatorial cyclic actions, provides an equivariant bijection between plane partitions and increasing tableaux, and generalizes promotion-rowmotion conjugacy.

Findings

Orbit sizes are multiples of divisors of a fundamental frequency.

K-promotion cyclically rotates labels in increasing tableaux.

New results on the order of K-promotion and resonance phenomena.

Abstract

We introduce a new concept of resonance on discrete dynamical systems. This concept formalizes the observation that, in various combinatorially-natural cyclic group actions, orbit cardinalities are all multiples of divisors of a fundamental 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 increasing tableaux under $K$-promotion. Both of these actions were observed to have orbit sizes that were small multiples of divisors of an expected orbit size, and we show this is an instance of resonance, as $K$-promotion cyclically rotates the set of labels appearing in the increasing tableaux. We extract a number of corollaries from this equivariant bijection, including a strengthening of a theorem of [P. Cameron--D. Fon-der-Flaass '95] and several new results on the order of $K$-promotion. Along the way, we adapt the proof of the conjugacy of promotion and rowmotion from [J. Striker--N. Williams '12] to give a generalization in the setting of $n$-dimensional lattice projections. Finally we discuss known and conjectured examples of resonance relating to alternating sign matrices and fully-packed loop configurations.