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.Comment: 31 pages, 12 figure
Abstract. An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.
Let G be a group acting on a set X of combinatorial objects, with finite orbits, and consider a statistic ξ : X → C. Propp and Roby defined the triple (X, G, ξ) to be homomesic if for any orbits O 1 , O 2 , the average value of the statistic ξ is the same, that isIn 2013 Propp and Roby conjectured the following instance of homomesy. Let SSYT k (m×n) denote the set of semistandard Young tableaux of shape m × n with entries bounded by k. Let S be any set of boxes in the m × n rectangle fixed under 180 • rotation. For T ∈ SSYT k (m × n), define σ S (T ) to be the sum of the entries of T in the boxes of S. Let P be a cyclic group of order k where P acts on SSYT k (m × n) by promotion. Then (SSYT k (m × n), P , σ S ) is homomesic.We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the Kpromotion of Thomas and Yong, and prove limited results in that direction.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.