Pop-Stack Operators for Torsion Classes and Cambrian Lattices

The pop-stack operator of a finite lattice 𝐿 is the map pop ↓ 𝐿 : 𝐿 → 𝐿 that sends each element 𝑥 ∈ 𝐿 to the meet of { 𝑥 } ∪ cov 𝐿 ( 𝑥 ), where cov 𝐿 ( 𝑥 ) is the set of elements covered by 𝑥 in 𝐿.We study several properties of the pop-stack operator of tors Λ, the lattice of torsion classes of a 𝜏-tilting finite algebra Λ 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.