Pop-Stack Operators for Torsion Classes and Cambrian Lattices

Abstract

The pop-stack operator of a finite lattice $L$ is the map ${\mathrm{pop}}_L^{\downarrow }:L\to L$ that sends each element $x\in L$ to the meet of $\{x\}\cup {\text{cov}}_L(x)$, where ${\text{cov}}_L(x)$ is the set of elements covered by $x$ in $L$. We study several properties of the pop-stack operator of $\mathrm{tors}\hat{I}›$, the lattice of torsion classes of a $\ddot{I}„$-tilting finite algebra $\hat{I}›$ over a field $K$. We describe the pop-stack operator in terms of certain mutations of 2-term simple-minded collections. This allows us to describe preimages of a given torsion class under the pop-stack operator. We then specialize our attention to Cambrian lattices of a finite irreducible Coxeter group $W$. Using tools from representation theory, we provide simple Coxeter-theoretic and lattice-theoretic descriptions of the image of the pop-stack operator of a Cambrian lattice (which can be stated without representation theory). When specialized to a bipartite Cambrian lattice of type A, this result settles a conjecture of Choi and Sun. We also settle a related enumerative conjecture of Defant and Williams. When $L$ is an arbitrary lattice quotient of the weak order on $W$, we prove that the maximum size of a forward orbit under the pop-stack operator of $L$ is at most the Coxeter number of $W$; when $L$ is a Cambrian lattice, we provide an explicit construction to show that this maximum forward orbit size is actually equal to the Coxeter number.

50 pages, 7 figures