Treillis de Cayley des groupes de Coxeter finis. Constructions par récurrence et décompositions sur des quotients

This article, dedicated to André Lentin on the occasionof the meeting (23/02/1996) organized in his honor, aims to show the the labelled lattice obtained from the weak order on a finite Coxeter system (W,S) as welle as the group itself can be constructed stating from an arbitrary parabolic subgroupe WJ, the associated quotient WJ and a function from WJ x J to S U . This method permits to recursive construction of groups and latices inthe four infinite families of irreducible finite Coxter groups : the reverse procedure leads to a reduction algorithm for expressions of elements of the group as products of generatorsCet article, offert à André Lentin lors du colloque du 23 février 1996 organisé en son honneur, a pour objet de montrer que le treillis étiqueté obtenu à partir de l'ordre faible sur un Coxeter fini (W,S), et le groupe lui-même peuvent être construits à partir d'un sous-groupe parabolique quelconque Wj, du quotient associé Wj et d'une fonction de Wj x Wj dans S appartenant à l'ensemble vide. Cette méthode permet en particulier la construction par récurrence des groupes et treillis des quatre familles infinies de Coxeter finis irréductibles et la procédure inverse, la réduction de toute décomposition des éléments du groupe

Cayley lattices of finite Coxeter groups are bounded

AbstractAn 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 LW and a parabolic subgroup WH of the finite Coxeter group W, we can construct LW starting from WH by a series of interval doublings. For instance the lattice, associated with An, of all the permutations on n+1 elements is obtained from An−1 by a series of interval doublings. The same holds for the lattices associated with the other infinite families of Coxeter groups Bn, Dn and I2(n)

The biHecke monoid of a finite Coxeter group and its representations

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H_0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w in W a combinatorial module T_w whose support is the interval [1,w]_R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.Comment: v2: Added complete description of the rank 2 case (Section 7.3) and improved proof of Proposition 7.5. v3: Final version (typo fixes, picture improvements) 66 pages, 9 figures Algebra and Number Theory, 2013. arXiv admin note: text overlap with arXiv:1108.4379 by other author