Séminaire Lotharingien de Combinatoire, 91B.30 (2024), 12 pp.
Emily Barnard, Colin Defant, and Eric J. Hanson
Pop-Stack for Cambrian Lattices
Abstract.
The pop-stack operator of a finite lattice L is the map that
sends each x in L to the meet of x with the set of elements
covered by x. 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 of a finite
irreducible Coxeter group. 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.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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