Resonance in orbits of plane partitions and increasing tableaux

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:148

Article

Dilks, Kevin¹  Pechenik, Oliver²  Striker, Jessica¹ 

[1] North Dakota State Univ, Dept Math, Fargo, ND 58102 USA

[2] Rutgers State Univ, Dept Math, Piscataway, NJ 08854 USA

关键词: Resonance;    Increasing tableau;    Plane partition;    Rowmotion;    Promotion;    K-theoretic jeu de taquin;    Alternating sign matrix;    Descent;    K-Bender-Knuth;   

DOI  :  10.1016/j.jcta.2016.12.007

来源: Elsevier

【 摘 要 】

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 Cameron and Fon-der-Flaass (1995) [9] 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 Striker and Williams (2012) [38] 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. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free