Crystal Pop-Stack Sorting and Type A Crystal Lattices

TL;DR

This paper introduces a new crystal pop-stack sorting operator on crystal posets associated with Lie algebras, analyzes its orbit structure, and characterizes when type A crystals form lattices, advancing understanding of crystal combinatorics.

Contribution

It defines the crystal pop-stack sorting operator, studies its orbit properties, and characterizes lattice conditions for type A crystals, extending sorting concepts to Lie algebra representations.

Findings

Every orbit contains the minimal fixed element.

Maximum orbit size equals the Coxeter number.

Characterization of lattice crystals in type A.

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.