1,066 research outputs found
Noncrossing partitions and the shard intersection order
We define a new lattice structure on the elements of a finite Coxeter group
W. This lattice, called the shard intersection order, is weaker than the weak
order and has the noncrossing partition lattice NC(W) as a sublattice. The new
construction of NC(W) yields a new proof that NC(W) is a lattice. The shard
intersection order is graded and its rank generating function is the W-Eulerian
polynomial. Many order-theoretic properties of the shard intersection order,
like Mobius number, number of maximal chains, etc., are exactly analogous to
the corresponding properties of NC(W). There is a natural dimension-preserving
bijection between simplices in the order complex of the shard intersection
order (i.e. chains in the shard intersection order) and simplices in a certain
pulling triangulation of the W-permutohedron. Restricting the bijection to the
order complex of NC(W) yields a bijection to simplices in a pulling
triangulation of the W-associahedron.
The shard intersection order is defined indirectly via the polyhedral
geometry of the reflecting hyperplanes of W. Indeed, most of the results of the
paper are proven in the more general setting of simplicial hyperplane
arrangements.Comment: 44 pages, 15 figure
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
We show that the order dimension of the weak order on a Coxeter group of type
A, B or D is equal to the rank of the Coxeter group, and give bounds on the
order dimensions for the other finite types. This result arises from a unified
approach which, in particular, leads to a simpler treatment of the previously
known cases, types A and B. The result for weak orders follows from an upper
bound on the dimension of the poset of regions of an arbitrary hyperplane
arrangement. In some cases, including the weak orders, the upper bound is the
chromatic number of a certain graph. For the weak orders, this graph has the
positive roots as its vertex set, and the edges are related to the pairwise
inner products of the roots.Comment: Minor changes, including a correction and an added figure in the
proof of Proposition 2.2. 19 pages, 6 figure
Lattice congruences of the weak order
We study the congruence lattice of the poset of regions of a hyperplane
arrangement, with particular emphasis on the weak order on a finite Coxeter
group. Our starting point is a theorem from a previous paper which gives a
geometric description of the poset of join-irreducibles of the congruence
lattice of the poset of regions in terms of certain polyhedral decompositions
of the hyperplanes. For a finite Coxeter system (W,S) and a subset K of S, let
\eta_K:w \mapsto w_K be the projection onto the parabolic subgroup W_K. We show
that the fibers of \eta_K constitute the smallest lattice congruence with
1\equiv s for every s\in(S-K). We give an algorithm for determining the
congruence lattice of the weak order for any finite Coxeter group and for a
finite Coxeter group of type A or B we define a directed graph on subsets or
signed subsets such that the transitive closure of the directed graph is the
poset of join-irreducibles of the congruence lattice of the weak order.Comment: 26 pages, 4 figure
Hyperplane Arrangements with Large Average Diameter
The largest possible average diameter of a bounded cell of a simple
hyperplane arrangement is conjectured to be not greater than the dimension. We
prove that this conjecture holds in dimension 2, and is asymptotically tight in
fixed dimension. We give the exact value of the largest possible average
diameter for all simple arrangements in dimension 2, for arrangements having at
most the dimension plus 2 hyperplanes, and for arrangements having 6
hyperplanes in dimension 3. In dimension 3, we give lower and upper bounds
which are both asymptotically equal to the dimension
Output-Sensitive Tools for Range Searching in Higher Dimensions
Let be a set of points in . A point is
\emph{-shallow} if it lies in a halfspace which contains at most points
of (including ). We show that if all points of are -shallow, then
can be partitioned into subsets, so that any hyperplane
crosses at most subsets. Given such
a partition, we can apply the standard construction of a spanning tree with
small crossing number within each subset, to obtain a spanning tree for the
point set , with crossing number . This allows us to extend the construction of Har-Peled
and Sharir \cite{hs11} to three and higher dimensions, to obtain, for any set
of points in (without the shallowness assumption), a
spanning tree with {\em small relative crossing number}. That is, any
hyperplane which contains points of on one side, crosses
edges of . Using a
similar mechanism, we also obtain a data structure for halfspace range
counting, which uses space (and somewhat higher
preprocessing cost), and answers a query in time , where is the output size
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