On the Order of P-Strict Promotion on $V\times [\ell]$

TL;DR

This paper proves that the order of P-strict promotion on a specific poset product is 2q, confirming a conjecture and resolving an equivalent conjecture about piecewise-linear rowmotion.

Contribution

It establishes the exact order of P-strict promotion on V×[ℓ], confirming a conjecture and linking it to rowmotion on the order polytope.

Findings

Order of P-strict promotion is 2q for all ℓ≥1 and q≥3.

Confirms the conjecture by Bernstein, Striker, and Vorland.

Resolves Hopkins' conjecture on rowmotion order.

Abstract

Denote by $V$ the poset consisting of the elements $\{A,B,C\}$ with cover relations $\{A\lessdot B, A\lessdot C\}$. We show that $P$-strict promotion, as defined by Bernstein, Striker, and Vorland, on $P$-strict labelings of $V\times [\ell]$ with labels in the set $[q]$ has order $2q$ for every $\ell\ge 1$ and $q\ge 3$ as conjectured by Bernstein, Striker, and Vorland. This resolves the equivalent conjecture of Hopkins that the order of piecewise-linear rowmotion on the order polytope of $V\times [k]$ has order $2(k+2)$ for all $k\ge 1$.