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arXiv · 2109.08251

Crystal Pop-Stack Sorting and Type A Crystal Lattices

Colin Defant·Nathan Williams
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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.

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Colin Defant, Nathan Williams. 2021-09-16. Crystal Pop-Stack Sorting and Type A Crystal Lattices. https://arxiv.org/abs/2109.08251

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