Cayley lattices of finite Coxeter groups are bounded

• Published in: Advances in applied mathematics 2004-07, Vol.33 (1), p.71-94
• Main Authors: Caspard, Nathalie, Le Conte de Poly-Barbut, Claude, Morvan, Michel
• Format: Article
• Language: English
• Subjects: Bounded lattice; Combinatorics. Ordered structures; Computer Science; Coxeter group and lattice; Discrete Mathematics; Doubling operation; Exact sciences and technology; Group theory; Group theory and generalizations; Mathematics; Order, lattices, ordered algebraic structures; Reflection; Sciences and techniques of general use; Semidistributivity; Zonotope
• ISSN: 0196-8858; 1090-2074
• DOI: 10.1016/j.aam.2003.09.002

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).