Crystal pop-stack sorting and type [formula omitted] crystal lattices

Abstract

Given a complex simple Lie algebra g and a dominant weight λ, let Bλ be the crystal poset associated to the irreducible representation of g with highest weight λ. In the first part of the article, we introduce the crystal pop-stack sorting operatorPop◊:Bλ→Bλ, 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 Pop◊ contains the minimal element of Bλ, which is fixed by Pop◊. We prove that the maximum size of a forward orbit of Pop◊ is the Coxeter number of the Weyl group of g. In the second part of the article, we characterize exactly when a type A crystal is a lattice.