257 research outputs found
On the Sperner property for the absolute order on complex reflection groups
Two partial orders on a reflection group, the codimension order and the
prefix order, are together called the absolute order when they agree. We show
that in this case the absolute order on a complex reflection group has the
strong Sperner property, except possibly for the Coxeter group of type ,
for which this property is conjectural. The Sperner property had previously
been established for the noncrossing partition lattice , a certain
maximal interval in the absolute order, but not for the entire poset, except in
the case of the symmetric group. We also show that neither the codimension
order nor the prefix order has the Sperner property for general complex
reflection groups.Comment: 12 pages, comments welcome; v2: minor edits and journal referenc
Counting Shellings of Complete Bipartite Graphs and Trees
A shelling of a graph, viewed as an abstract simplicial complex that is pure
of dimension 1, is an ordering of its edges such that every edge is adjacent to
some other edges appeared previously. In this paper, we focus on complete
bipartite graphs and trees. For complete bipartite graphs, we obtain an exact
formula for their shelling numbers. And for trees, we propose a simple method
to count shellings and bound shelling numbers using vertex degrees and
diameter.Comment: 22 pages, 6 figure
Balance constants for Coxeter groups
The - Conjecture, originally formulated in 1968, is one of the
best-known open problems in the theory of posets, stating that the balance
constant (a quantity determined by the linear extensions) of any non-total
order is at least . By reinterpreting balance constants of posets in terms
of convex subsets of the symmetric group, we extend the study of balance
constants to convex subsets of any Coxeter group. Remarkably, we conjecture
that the lower bound of still applies in any finite Weyl group, with new
and interesting equality cases appearing.
We generalize several of the main results towards the - Conjecture
to this new setting: we prove our conjecture when is a weak order interval
below a fully commutative element in any acyclic Coxeter group (an
generalization of the case of width-two posets), we give a uniform lower bound
for balance constants in all finite Weyl groups using a new generalization of
order polytopes to this context, and we introduce generalized semiorders for
which we resolve the conjecture.
We hope this new perspective may shed light on the proper level of generality
in which to consider the - Conjecture, and therefore on which methods
are likely to be successful in resolving it.Comment: 27 page
Quantum Bruhat graph and tilted Richardson varieties
Quantum Bruhat graph is a weighted directed graph on a finite Weyl group
first defined by Brenti-Fomin-Postnikov. It encodes quantum Monk's rule and can
be utilized to study the -point Gromov-Witten invariants of the flag
variety. In this paper, we provide an explicit formula for the minimal weights
between any pair of permutations on the quantum Bruhat graph, and consequently
obtain an Ehresmann-like characterization for the tilted Bruhat order.
Moreover, for any ordered pair of permutations and , we define the
tilted Richardson variety , with a stratification that gives a
geometric meaning to intervals in the tilted Bruhat order. We provide a few
equivalent definitions to this new family of varieties that include Richardson
varieties, and establish some fundamental geometric properties including their
dimensions and closure relations.Comment: 28 page
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