Rowmotion on the chain of V's poset and whirling dynamics

TL;DR

This paper explores the whirling action on labelings of a poset, establishing a bijection to order ideals, and applies this to derive periodicity and homomesy results for rowmotion on specific posets, including the chain of V's and claw posets.

Contribution

It introduces an equivariant bijection linking whirling on bounded P-partitions to rowmotion, enabling new results on periodicity and homomesy in these dynamics.

Findings

Established a bijection between P-partitions and order ideals of P×[k].

Derived periodicity results for rowmotion on V×[k].

Extended some results to rowmotion on C_{n}×[k].

Abstract

Given a finite poset $P$, we study the _whirling_ action on vertex-labelings of $P$ with the elements $\{0,1,2,\dotsc ,k\}$. When such labelings are (weakly) order-reversing, we call them $k$-bounded $P$-partitions. We give a general equivariant bijection between $k$-bounded $P$-partitions and order ideals of the poset $P\times [k]$ which conveys whirling to the well-studied rowmotion operator. As an application, we derive periodicity and homomesy results for rowmotion acting on the chain of V's poset $V \times [k]$. We are able to generalize some of these results to the more complicated dynamics of rowmotion on $C_{n}\times [k]$, where $C_{n}$ is the claw poset with $n$ unrelated elements each covering $\widehat{0}$.