Cayley lattices of finite Coxeter groups are bounded

Abstract

An interval doubling is a constructive operation which applies on a poset P and an interval I of P and constructs a new "bigger" poset P = P [I ] by replacing in P the interval I with its direct product with the two-element lattice. The main contribution of this paper is to prove that finite Coxeter lattices are bounded, i.e., that they can be constructed starting with the two-element lattice by a finite series of interval doublings.The boundedness of finite Coxeter lattices strengthens their algebraic property of semidistributivity. It also brings to light a relation between the interval doubling construction and the reflections of Coxeter groups.Our approach to the question is somewhat indirect. We first define a new class HH of lattices and prove that every lattice of HH is bounded. We then show that Coxeter lattices are in HH and the theorem follows. Another result says that, given a Coxeter lattice L W and a parabolic subgroup W H of the finite Coxeter group W , we can construct L W starting from W H by a series of interval doublings. For instance the lattice, associated with A n , of all the permutations on n + 1 elements is obtained from A n−1 by a series of interval doublings. The same holds for the lattices associated with the other infinite families of Coxeter groups B n , D n and I 2 (n).

KP line solitons and Tamari lattices

“…As indicated by equally encircled parts, the deformation of a class into the next, as t (4) passes through a critical value, proceeds according to the left to right half pentagon structure of T 3 . [40].) Here the numbers assigned to the pentagons encode the critical times associated with the edges.…”

Derived semidistributive lattices

“…We introduce them now together with their elementary properties. Recall from [5] that a hat in a poset P is a triple (u, v, w) such that u ≺ v, w ≺ v, and u = w. An antihat in P is defined dually. A cospan in P is a triple of elements (u, v, w) such that u ≤ v and w ≤ v; in particular, a hat is a particular kind of a cospan.…”

“…The latter property may be rephrased by saying that for each γ ∈ C(L) there exists a unique ι ∈ C(L), minimal within C(L), such that ι and γ are comparable. Following a suggestion of [5,Theorem 1], we observe that such uniqueness property is consequence of the pushdown relation between covers being confluent. (See for example [16, §3] or [8, §4].)…”

“…On the side of lower bounded lattices, we build on the ideas used in [5] to prove that lattices arising from Cayley graphs of finite Coxeter groups are bounded. With respect to that work, we move from a sufficient condition to a complete characterization, and from boundedness to the weaker notion of lower boundedness.…”

“…As illustrated in Figure 1, a facet is a quadruple of covers δ, δ , γ, γ ∈ C(L) such that γ 0 = γ 0 = δ 0 ∧ δ 0 < γ 1 ∨ γ 1 = δ 1 = δ 1 , with γ 1 ≤ δ 0 and γ 1 ≤ δ 0 . (In [5] a facet is called a 2-facet, using a more precise wording from combinatorial geometry.) For such a facet, we shall prove that γ ≺ δ (as well as γ ≺ δ ) in the poset C(L) and, moreover, that every cover of covers arises from a facet.…”