Crystal Pop-Stack Sorting and Type A Crystal Lattices

Abstract

Given a complex simple Lie algebra $\mathfrak{g}$ and a dominant weight $\hat{I}»$, let ${\mathcal{B}}_{\hat{I}}»$ be the crystal poset associated to the irreducible representation of $\mathfrak{g}$ with highest weight $\hat{I}»$. In the first part of the article, we introduce the \emph{crystal pop-stack sorting operator} ${\mathsf{Pop}}_{◊}:{\mathcal{B}}_{\hat{I}}»\to {\mathcal{B}}_{\hat{I}}»$, 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 ${\mathsf{Pop}}_{◊}$ contains the minimal element of ${\mathcal{B}}_{\hat{I}}»$, which is fixed by ${\mathsf{Pop}}_{◊}$. We prove that the maximum size of a forward orbit of ${\mathsf{Pop}}_{◊}$ 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.

16 pages, 4 figures