2023/12/07 by Barnard, Emily, Defant, Colin, Hanson, Eric J. · 1 citation
#05A05 #05A15 #05A19 #05E10 #05E16 #06D75 #06F15 #16G10 #16G20 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2312.03959
The pop-stack operator of a finite lattice L is the map pop\downarrowL\colon L→ L that sends each element x∈ L to the meet of \x\\cupcovL(x), where covL(x) is the set of elements covered by x in L. We study several properties of the pop-stack operator of torsΛ, 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.