Polygon Posets and the Weak Order of Coxeter Groups
ثبت نشده
چکیده
We explore the connection between polygon posets, which is a class of ranked posets with an edgelabeling which satisfies certain 'polygon properties', and the weak order of Coxeter groups. We show that every polygon poset is isomorphic to a join ideal in the weak order, and for Coxeter groups where no pair of generators have infinite order the converse is also true. The class of polygon posets is seen to include the class of generalized quotients defined by Bjorner and Wachs, while itself being included in the class of alternative generalized quotients also considered by these authors. By studying polygon posets we are then able to answer an open question about common properties of these two classes.
منابع مشابه
Order dense injectivity of $S$-posets
In this paper, the notion of injectivity with respect to order dense embeddings in the category of $S$-posets, posets with a monotone action of a pomonoid $S$ on them, is studied. We give a criterion, like the Baer condition for injectivity of modules, or Skornjakov criterion for injectivity of $S$-sets, for the order dense injectivity. Also, we consider such injectivit...
متن کاملOn the Coxeter transformations for Tamari posets
A relation between the anticyclic structure of the dendriform operad and the Coxeter transformations in the Grothendieck groups of the derived categories of modules over the Tamari posets is obtained.
متن کاملAlgebraic and combinatorial properties of zircons
In this paper we introduce and study a new class of posets, that we call zircons, which includes all Coxeter groups partially ordered by Bruhat order. We prove that many of the properties of Coxeter groups extend to zircons often with simpler proofs: in particular, zircons are Eulerian posets and the Kazhdan-Lusztig construction of the Kazhdan-Lusztig representations can be carried out in the c...
متن کاملThe Combinatorics of Reduced Decompositions
This thesis examines several aspects of reduced decompositions in finite Coxeter groups. Effort is primarily concentrated on the symmetric group, although some discussions are subsequently expanded to finite Coxeter groups of types B and D. In the symmetric group, the combined frameworks of permutation patterns and reduced decompositions are used to prove a new characterization of vexillary per...
متن کاملVarious kinds of regular injectivity for $S$-posets
In this paper some properties of weak regular injectivity for $S$-posets, where $S$ is a pomonoid, are studied. The behaviour of different kinds of weak regular injectivity with products, coproducts and direct sums is considered. Also, some characterizations of pomonoids over which all $S$-posets are of some kind of weakly regular injective are obtained. Further, we giv...
متن کامل