668 research outputs found
The sorting order on a Coxeter group
Let be an arbitrary Coxeter system. For each word in the
generators we define a partial order--called the {\sf -sorting
order}--on the set of group elements that occur as
subwords of . We show that the -sorting order is a
supersolvable join-distributive lattice and that it is strictly between the
weak and Bruhat orders on the group. Moreover, the -sorting order is a
"maximal lattice" in the sense that the addition of any collection of Bruhat
covers results in a nonlattice. Along the way we define a class of structures
called {\sf supersolvable antimatroids} and we show that these are equivalent
to the class of supersolvable join-distributive lattices.Comment: 34 pages, 7 figures. Final version, to appear in Journal of
Combinatorial Theory Series
Stack-Sorting for Coxeter Groups
Given an essential semilattice congruence on the left weak order of
a Coxeter group , we define the Coxeter stack-sorting operator by , where
is the unique minimal element of the congruence
class of containing . When is the sylvester congruence on
the symmetric group , the operator is West's
stack-sorting map. When is the descent congruence on , the
operator is the pop-stack-sorting map. We establish several
general results about Coxeter stack-sorting operators, especially those acting
on symmetric groups. For example, we prove that if is an essential
lattice congruence on , then every permutation in the image of has at most right
descents; we also show that this bound is tight.
We then introduce analogues of permutree congruences in types and
and use them to isolate Coxeter stack-sorting operators
and that serve as
canonical type- and type- counterparts of West's stack-sorting
map. We prove analogues of many known results about West's stack-sorting map
for the new operators and
. For example, in type , we
obtain an analogue of Zeilberger's classical formula for the number of
-stack-sortable permutations in .Comment: 39 pages, 11 figure
On the Topology of the Cambrian Semilattices
For an arbitrary Coxeter group , David Speyer and Nathan Reading defined
Cambrian semilattices as semilattice quotients of the weak order
on induced by certain semilattice homomorphisms. In this article, we define
an edge-labeling using the realization of Cambrian semilattices in terms of
-sortable elements, and show that this is an EL-labeling for every
closed interval of . In addition, we use our labeling to show that
every finite open interval in a Cambrian semilattice is either contractible or
spherical, and we characterize the spherical intervals, generalizing a result
by Nathan Reading.Comment: 20 pages, 5 figure
Shelling Coxeter-like Complexes and Sorting on Trees
In their work on `Coxeter-like complexes', Babson and Reiner introduced a
simplicial complex associated to each tree on nodes,
generalizing chessboard complexes and type A Coxeter complexes. They
conjectured that is -connected when the tree has
leaves. We provide a shelling for the -skeleton of , thereby
proving this conjecture.
In the process, we introduce notions of weak order and inversion functions on
the labellings of a tree which imply shellability of , and we
construct such inversion functions for a large enough class of trees to deduce
the aforementioned conjecture and also recover the shellability of chessboard
complexes with . We also prove that the existence or
nonexistence of an inversion function for a fixed tree governs which networks
with a tree structure admit greedy sorting algorithms by inversion elimination
and provide an inversion function for trees where each vertex has capacity at
least its degree minus one.Comment: 23 page
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
Trimness of Closed Intervals in Cambrian Semilattices
In this article, we give a short algebraic proof that all closed intervals in
a -Cambrian semilattice are trim for any Coxeter
group and any Coxeter element . This means that if such an
interval has length , then there exists a maximal chain of length
consisting of left-modular elements, and there are precisely join- and
meet-irreducible elements in this interval. Consequently every graded interval
in is distributive. This problem was open for any
Coxeter group that is not a Weyl group.Comment: Final version. The contents of this paper were formerly part of my
now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure
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