The sorting order on a Coxeter group

@article{Armstrong2009TheSO,
  title={The sorting order on a Coxeter group},
  author={Drew Armstrong},
  journal={J. Comb. Theory, Ser. A},
  year={2009},
  volume={116},
  pages={1285-1305}
}
  • D. Armstrong
  • Published 2009
  • Computer Science, Mathematics
  • J. Comb. Theory, Ser. A
Let (W,S) be an arbitrary Coxeter system. For each word @w in the generators we define a partial order-called the @w-sorting order-on the set of group elements W"@w@?W that occur as subwords of @w. We show that the @w-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and Bruhat orders on the group. Moreover, the @w-sorting order is a ''maximal lattice'' in the sense that the addition of any collection of Bruhat covers results in a nonlattice… Expand

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References

SHOWING 1-10 OF 29 REFERENCES
Sortable elements and Cambrian lattices
Abstract.We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify thisExpand
Subword complexes in Coxeter groups
Abstract Let ( Π , Σ ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex , as introduced in Knutson and Miller (Ann. of Math. (2) (2003), toExpand
Meet-distributive lattices and the anti-exchange closure
This paper defines the anti-exchange closure, a generalization of the order ideals of a partially ordered set. Various theorems are proved about this closure. The main theorem presented is that aExpand
Powers of Coxeter elements in infinite groups are reduced
Let W be an infinite irreducible Coxeter group with (s 1 ,..., s n ) the simple generators. We give a short proof that the word s 1 s 2 ... s n s 1 s 2 ··· s n ···s 1 s 2 ...s n is reduced for anyExpand
Join-semidistributive lattices and convex geometries
Abstract We introduce the notion of a convex geometry extending the notion of a finite closure system with the anti-exchange property known in combinatorics. This notion becomes essential for theExpand
The lattice of closure relations on a poset
In this paper we show that the set of closure relations on a finite posetP forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice is dually isomorphic to the lattice of closedExpand
Clusters, Coxeter-sortable elements and noncrossing partitions
We introduce Coxeter-sortable elements of a Coxeter group For finite we give bijective proofs that Coxeter-sortable elements are equinumerous with clusters and with noncrossing partitions. WeExpand
What Is Enumerative Combinatorics
The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually are given an infinite class of finite sets S i where i ranges over some index set IExpand
Combinatorics of Coxeter Groups
I.- The basics.- Bruhat order.- Weak order and reduced words.- Roots, games, and automata.- II.- Kazhdan-Lusztig and R-polynomials.- Kazhdan-Lusztig representations.- Enumeration.- CombinatorialExpand
Lattices with unique irreducible decompositions
We define the concept of a minimal decomposition in a lattice, and prove that all the currently known lattices with unique irreducible decompositions are in fact lattices with minimal ones. Also, theExpand
...
1
2
3
...