66 research outputs found
Dimension of CPT posets
A collection of linear orders on , say , is said to
\emph{realize} a partially ordered set (or poset)
if, for any two distinct , if and only if , . We call a \emph{realizer} of
. The \emph{dimension} of , denoted by
, is the minimum cardinality of a realizer of .
A \emph{containment model} of a poset
maps every to a set such that, for
every distinct if and only if . We shall be using the collection to identify the
containment model . A poset is a
Containment order of Paths in a Tree (CPT poset), if it admits a containment
model where every is a path of a tree
, which is called the host tree of the model.
We show that if a poset admits a CPT model in a host tree
of maximum degree and radius , then \rogers{. This bound is asymptotically tight up to an
additive factor of .
Further, let be the poset consisting of all the
-element and -element subsets of under `containment' relation and
let denote its dimension. The proof of our main theorem gives a
simple algorithm to construct a realizer for whose
cardinality is only an additive factor of at most away from the
optimum.Comment: 10 Page
Homology of generalized partition posets
We define a poset of partitions associated to an operad. We prove that the
operad is Koszul if and only if the poset is Cohen-Macaulay.
In one hand, this characterisation allows us to compute the homology of the
poset. This homology is given by the Koszul dual operad. On the other hand, we
get new methods for proving that an operad is Koszul.Comment: Final version. To appear in JPA
Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group
This paper extends certain characterizations of nullhomotopic maps between
p-compact groups to maps with target the p-completed classifying space of a
connected Kac-Moody group and source the classifying space of either a
p-compact group or a connected Kac-Moody group. A well known inductive
principle for p-compact groups is applied to obtain general, mapping space
level results. An arithmetic fiber square computation shows that a null map
from the classifying space of a connected compact Lie group to the classifying
space of a connected topological Kac-Moody group can be detected by restricting
to the maximal torus. Null maps between the classifying spaces of connected
topological Kac-Moody groups cannot, in general, be detected by restricting to
the maximal torus due to the nonvanishing of an explicit abelian group of
obstructions described here. Nevertheless, partial results are obtained via the
application of algebraic discrete Morse theory to higher derived limit
calculations which show that such detection is possible in many cases of
interest.Comment: References added, minor corrections; 29 pages, 4 figures, one tabl
Moduli stack of stable curves from a stratified homotopy viewpoint
In 1984, Charney and Lee defined a category of stable curves and exhibited a
rational homology equivalence from its geometric realisation to (the
analytification of) the moduli stack of stable curves, also known as the
Deligne-Mumford-Knudsen compactification. We strengthen this result by showing
that, in fact, this category captures the stratified homotopy type of the
moduli stack. In particular, it classifies constructible sheaves via an
exodromy equivalence.Comment: 55 pages (including a 15 page appendix on the Harvey
compactification), 3 figures. Comments welcome; v2 minor change
Homomesy via Toggleability Statistics
The rowmotion operator acting on the set of order ideals of a finite poset
has been the focus of a significant amount of recent research. One of the major
goals has been to exhibit homomesies: statistics that have the same average
along every orbit of the action. We systematize a technique for proving that
various statistics of interest are homomesic by writing these statistics as
linear combinations of "toggleability statistics" (originally introduced by
Striker) plus a constant. We show that this technique recaptures most of the
known homomesies for the posets on which rowmotion has been most studied. We
also show that the technique continues to work in modified contexts. For
instance, this technique also yields homomesies for the piecewise-linear and
birational extensions of rowmotion; furthermore, we introduce a -analogue of
rowmotion and show that the technique yields homomesies for "-rowmotion" as
well.Comment: 48 pages, 13 figures, 2 tables; forthcoming, Combinatorial Theor
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