Crystal Pop-Stack Sorting and Type A Crystal Lattices

Given a complex simple Lie algebra $\mathfrak{g}$ and a dominant weight $\lambda$, let ${\mathcal{B}}_{\lambda }$ be the crystal poset associated to the irreducible representation of $\mathfrak{g}$ with highest weight $\lambda$. In the first part of the article, we introduce the \emph{crystal pop-stack sorting operator} $$\mathsf{P}\mathsf{o}\mathsf{p}${\lozenge}\colon\mathcal B\lambda\to\mathcal B_\lambda$, a noninvertible operator whose definition extends that of the pop-stack sorting map and the recently-introduced Coxeter pop-stack sorting operators. Every forward orbit of ${\mathsf{P}\mathsf{o}\mathsf{p}}_{\mathrm{◊}}$ contains the minimal element of ${\mathcal{B}}_{\lambda }$, which is fixed by ${\mathsf{P}\mathsf{o}\mathsf{p}}_{\mathrm{◊}}$. We prove that the maximum size of a forward orbit of ${\mathsf{P}\mathsf{o}\mathsf{p}}_{\mathrm{◊}}$ is the Coxeter number of the Weyl group of $\mathfrak g$\mathrm{.}Inthesecondpartofthearticle,wecharacterizeexactlywhenatype$A$ crystal is a lattice.