Year: 2026
Type: article
Abstract: 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.
Source: Canadian Journal of Mathematics
Authors: Emily Barnard, Colin Defant, Eric J. Hanson
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W4389500954