Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance

Abstract

Mathematical inquiry often begins with the study of objects (numbers, shapes, variables, matrices, ideals, metric spaces, …) and the question, "What are the objects like?" It then moves to the study of actions (functions, rotations, reflections, multiplication, derivatives, shifts, …) and the question, "How do the objects behave?" The study of actions in various mathematical contexts has been extremely fruitful; consider the study of metric spaces through the lens of dynamical systems or the study of symmetries arising from group actions. For objects and actions arising from algebraic combinatorics, we call this study dynamical algebraic combinatorics.Let be a bijective action on a finite set . Such an action breaks the space into orbits. Often, the study of the orbits of provides insight into the structure of the objects in , revealing hidden symmetries and connections. One typically first seeks to understand the order of the action and then finds interesting properties the action exhibits. One surprisingly ubiquitous property is the cyclic sieving phenomenon [ReStWh04], which occurs when the evaluation of a generating function for at the primitive th root of unity (where = 2 / ) counts the number of elements of fixed under .For example, let be the set of binary words composed of two zeros and two ones. Let be the cyclic shift that acts by moving the first digit to the end of the word. Each word has four digits, so is of order four. Consider the inversion number statistic inv( ) of ∈ , which equals the number of pairs ( , ) with < such that the th digit of is 1 and the th digit is 0. See Figure 1 for the orbits of under and the inversion numbers of each binary word.

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

“…In recent years, especially in the context of "dynamical algebraic combinatorics" [68,86], other aspects of rowmotion beyond its orbit structure have been investigated. There has been a particular focus on exhibiting homomesies for rowmotion.…”

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

“…Although Panyushev was not the first to consider rowmotion (which is defined more generally as acting on the antichains of any poset), his investigation of rowmotion on root posets rekindled interest in this operator. Indeed, the past 10 or so years have seen the emergence of the subfield of dynamical algebraic combinatorics [27,31], in which rowmotion features prominently. Furthermore, Panyushev's observation of 'constant average cardinality along orbits' was one of the first instances of homomesy [23], a phenomenon that again is at the heart of dynamical algebraic combinatorics.…”

$P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors

“…Rowmotion is an intriguing action that has recently generated significant interest as a prototypical action in dynamical algebraic combinatorics; see, for example, the survey articles [29,36]. Rowmotion was originally defined on hypergraphs by P. Duchet [11] and generalized to order ideals J(Q) of an arbitrary finite poset (Q, Q ) by A. Brouwer and A. Schrijver [4].…”