Abstract:

Dynamical algebraic combinatorics is a research area in combinatorics that studies phenomena arising when an action is repeatedly applied to a set of combinatorial objects, such as tableaux, order ideals of finite posets, words, and more. One of the most well-studied actions in this field is rowmotion. While the behavior of rowmotion is generally unpredictable, several families of posets are known to exhibit particularly nice dynamical properties under this action. In this talk, we introduce a new such family: ordinal sums of antichains.

The talk will begin with an introduction to rowmotion at various levels: combinatorial, plane partition, piecewise-linear, and birational. We will then demonstrate that the dynamics of plane partition rowmotion on ordinal sums of antichains can be effectively reduced to those of combinatorial rowmotion. Using this reduction, we construct explicit maps between ordinal sums of antichains with isomorphic comparability graphs and prove part of a conjecture posed by Hopkins (2020). Specifically, we show that the posets \Lambda_{\mathbb{Q}^{2n}} and \Phi^+(I_2(2n)) share the same orbit structure under plane partition rowmotion, while also preserving the sum of so-called down-degree statistics on each orbit.

Moreover, we show that our maps lift naturally to the birational level via de-tropicalization. As a culminating result, we establish the following theorem: for ordinal sums of antichains, the following three notions are equivalent—(1) isomorphism of comparability graphs, (2) coincidence of rowmotion orbit structures at all levels, and (3) equality of order polynomials.

If time permits, we will conclude with several conjectures that extend or relate to our results.