Resonance in orbits of plane partitions and increasing tableaux

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.

31 pages, 12 figures

References in corpus (3)

• Equivariant K-theory of Grassmannians
• On Orbits of Order Ideals of Minuscule Posets II: Homomesy
• Increasing tableaux and Narayana numbers

Cited by in corpus (16)

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• Rowmotion in slow motion
• Rowmotion and Increasing Labeling Promotion
• $P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors
• Doppelgängers: Bijections of Plane Partitions
• Promotion of increasing tableaux: frames and homomesies
• A web basis of invariant polynomials from noncrossing partitions
• Rowmotion and generalized toggle groups
• A birational lifting of the Stanley-Thomas word on products of two chains
• Orbits of Plane Partitions of Exceptional Lie Type
• The genomic Schur function is fundamental-positive
• Dynamics of plane partitions: Proof of the Cameron-Fon-Der-Flaass conjecture
• Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
• Curious cyclic sieving on increasing tableaux
• Order polynomial product formulas and poset dynamics
• $P$-strict promotion and $Q$-partition rowmotion: the graded case