52 research outputs found
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now withdrawn submission arXiv:1312.4449. 12 pages, 3 figure
On Reflection Orders Compatible with a Coxeter Element
In this article we give a simple, almost uniform proof that the lattice of
noncrossing partitions associated with a well-generated complex reflection
group is lexicographically shellable. So far a uniform proof is available only
for Coxeter groups. In particular we show that, for any complex reflection
group and any element , every -compatible reflection order is a
recursive atom order of the corresponding interval in absolute order. Since any
Coxeter element in any well-generated complex reflection group admits
a -compatible reflection order, the lexicographic shellability follows
from a well-known result due to Bj\"orner and Wachs.Comment: This article was withdrawn, since the generalized statement that any
compatible order below some reflection group element in absolute order is a
recursive atom order is wrong. A counterexample is for instance the absolute
order interval between the identity and the longest element in . The
statement for Coxeter elements is probably true. Comments welcom
Distributive Lattices have the Intersection Property
Distributive lattices form an important, well-behaved class of lattices. They
are instances of two larger classes of lattices: congruence-uniform and
semidistributive lattices. Congruence-uniform lattices allow for a remarkable
second order of their elements: the core label order; semidistributive lattices
naturally possess an associated flag simplicial complex: the canonical join
complex. In this article we present a characterization of finite distributive
lattices in terms of the core label order and the canonical join complex, and
we show that the core label order of a finite distributive lattice is always a
meet-semilattice.Comment: 9 pages, 3 figures. Final version. Comments are very welcom
Symmetric Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
We prove that the noncrossing partition lattices associated with the complex
reflection groups for admit symmetric decompositions
into Boolean subposets. As a result, these lattices have the strong Sperner
property and their rank-generating polynomials are symmetric, unimodal, and
-nonnegative. We use computer computations to complete the proof that
every noncrossing partition lattice associated with a well-generated complex
reflection group is strongly Sperner, thus answering affirmatively a question
raised by D. Armstrong.Comment: 30 pages, 5 figures, 1 table. Final version. The results of the
initial version were extended to symmetric Boolean decompositions of
noncrossing partition lattice
A Heyting Algebra on Dyck Paths of Type and
In this article we investigate the lattices of Dyck paths of type and
under dominance order, and explicitly describe their Heyting algebra structure.
This means that each Dyck path of either type has a relative pseudocomplement
with respect to some other Dyck path of the same type. While the proof that
this lattice forms a Heyting algebra is quite straightforward, the explicit
computation of the relative pseudocomplements using the lattice-theoretic
definition is quite tedious. We give a combinatorial description of the Heyting
algebra operations join, meet, and relative pseudocomplement in terms of height
sequences, and we use these results to derive formulas for pseudocomplements
and to characterize the regular elements in these lattices.Comment: Final version. 21 pages, 5 figure
Structural Properties of the Cambrian Semilattices -- Consequences of Semidistributivity
The -Cambrian semilattices defined by Reading
and Speyer are a family of meet-semilattices associated with a Coxeter group
and a Coxeter element , and they are lattices if and only if
is finite. In the case where is the symmetric group
and is the long cycle the corresponding
-Cambrian lattice is isomorphic to the well-known Tamari lattice
. Recently, Kallipoliti and the author have investigated
from a topological viewpoint, and showed that many
properties of the Tamari lattices can be generalized nicely. In the present
article this investigation is continued on a structural level using the
observation of Reading and Speyer that is
semidistributive. First we prove that every closed interval of
is a bounded-homomorphic image of a free lattice (in
fact it is a so-called -lattice). Subsequently we prove that
each closed interval of is trim, we determine its
breadth, and we characterize the closed intervals that are dismantlable.Comment: This paper has been withdrawn by the author due to a gap in the proof
of Theorem 1.1(i). The results in Theorems 1.1(ii)-(iv) and 1.2, and those
needed for their proofs remain true, and will be addressed in separate
articles. I suspect that the claim of Theorem 1.1(i) is still true. In fact,
I suspect that quotients of HH-lattices are HH-lattices again. Comments are
very welcom
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