239 research outputs found
Noncrossing partitions, clusters and the Coxeter plane
When W is a finite Coxeter group of classical type (A, B, or D), noncrossing
partitions associated to W and compatibility of almost positive roots in the
associated root system are known to be modeled by certain planar diagrams. We
show how the classical-type constructions of planar diagrams arise uniformly
from projections of small W-orbits to the Coxeter plane. When the construction
is applied beyond the classical cases, simple criteria are apparent for
noncrossing and for compatibility for W of types H_3 and I_2(m) and less simple
criteria can be found for compatibility in types E_6, F_4 and H_4. Our
construction also explains why simple combinatorial models are elusive in the
larger exceptional types.Comment: Very minor changes, as suggested by the referee. This is essentially
the final version, which will appear in Sem. Lothar. Combin. 32 pages. About
12 of the pages are taken up by 29 figure
Dual braid monoids, Mikado braids and positivity in Hecke algebras
We study the rational permutation braids, that is the elements of an
Artin-Tits group of spherical type which can be written where
and are prefixes of the Garside element of the braid monoid. We give a
geometric characterization of these braids in type and and then
show that in spherical types different from the simple elements of the
dual braid monoid (for arbitrary choice of Coxeter element) embedded in the
braid group are rational permutation braids (we conjecture this to hold also in
type ).This property implies positivity properties of the polynomials
arising in the linear expansion of their images in the Iwahori-Hecke algebra
when expressed in the Kazhdan-Lusztig basis. In type , it implies
positivity properties of their images in the Temperley-Lieb algebra when
expressed in the diagram basis.Comment: 26 pages, 8 figure
Random 3-noncrossing partitions
In this paper, we introduce polynomial time algorithms that generate random
3-noncrossing partitions and 2-regular, 3-noncrossing partitions with uniform
probability. A 3-noncrossing partition does not contain any three mutually
crossing arcs in its canonical representation and is 2-regular if the latter
does not contain arcs of the form . Using a bijection of Chen {\it et
al.} \cite{Chen,Reidys:08tan}, we interpret 3-noncrossing partitions and
2-regular, 3-noncrossing partitions as restricted generalized vacillating
tableaux. Furthermore, we interpret the tableaux as sampling paths of
Markov-processes over shapes and derive their transition probabilities.Comment: 17 pages, 7 figure
On bi-free De Finetti theorems
We investigate possible generalizations of the de Finetti theorem to bi-free
probability. We first introduce a twisted action of the quantum permutation
groups corresponding to the combinatorics of bi-freeness. We then study
properties of families of pairs of variables which are invariant under this
action, both in the bi-noncommutative setting and in the usual noncommutative
setting. We do not have a completely satisfying analogue of the de Finetti
theorem, but we have partial results leading the way. We end with suggestions
concerning the symmetries of a potential notion of n-freeness.Comment: 16 pages. Major rewriting. In the first version the main theorem was
stated through an embedding into a B-B-noncommutative probability space
making it much weaker than what the proof really contains. It has therefore
been split into two independent statements clarifying how far we are able to
extend the de Finetti theorem to the bi-free settin
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