Crystal Pop-Stack Sorting and Type A Crystal Lattices

Abstract: 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} $\mathsf{Pop}{\lozenge}\colon\mathcal B\lambda\to\mathcal B_\lambda$, 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 $\mathsf{Pop}{\lozenge}$ contains the minimal element of $\mathcal B\lambda$, which is fixed by $\mathsf{Pop}{\lozenge}$. We prove that the maximum size of a forward orbit of $\mathsf{Pop}{\lozenge}$ is the Coxeter number of the Weyl group of $\mathfrak g$. In the second part of the article, we characterize exactly when a type $A$ crystal is a lattice.