710 research outputs found
On the weak order of Coxeter groups
This paper provides some evidence for conjectural relations between
extensions of (right) weak order on Coxeter groups, closure operators on root
systems, and Bruhat order. The conjecture focused upon here refines an earlier
question as to whether the set of initial sections of reflection orders,
ordered by inclusion, forms a complete lattice. Meet and join in weak order are
described in terms of a suitable closure operator. Galois connections are
defined from the power set of W to itself, under which maximal subgroups of
certain groupoids correspond to certain complete meet subsemilattices of weak
order. An analogue of weak order for standard parabolic subsets of any rank of
the root system is defined, reducing to the usual weak order in rank zero, and
having some analogous properties in rank one (and conjecturally in general).Comment: 37 pages, submitte
On rigidity of abstract root systems of Coxeter systems
We introduce and study a combinatorially defined notion of root basis of a
(real) root system of a possibly infinite Coxeter group. Known results on
conjugacy up to sign of root bases of certain irreducible finite rank real root
systems are extended to abstract root bases, to a larger class of real root
systems, and, with a short list of (genuine) exceptions, to infinite rank
irreducible Coxeter systems.Comment: 34 page
Semidirect product decomposition of Coxeter groups
Let be a Coxeter system, let be a partition of
such that no element of is conjugate to an element of , let
be the set of -conjugates of elements of and let
be the subgroup of generated by . We show
that and that is
a Coxeter system.Comment: 28 pages, one table. We have added some comments on parabolic
subgroups, double cosets representatives, finite and affine Weyl groups,
invariant theory, Solomon descent algebr
The nature of the observed free-electron-like state in a PTCDA monolayer on Ag(111)
A free-electron like band has recently been observed in a monolayer of PTCDA
(3,4,9,10-perylene tetracarboxylic dianhydride) molecules on Ag(111) by
two-photon photoemission [Schwalb et al., Phys. Rev. Lett. 101, 146801 (2008)]
and scanning tunneling spectroscopy [Temirov et al., Nature 444, 350 (2006)].
Using density functional theory calculations, we find that the observed
free-electron like band originates from the Shockley surface state band being
dramatically shifted up in energy by the interaction with the adsorbed
molecules while it acquires also a substantial admixture with a molecular band
Imaginary cones and limit roots of infinite Coxeter groups
Let (W,S) be an infinite Coxeter system. To each geometric representation of
W is associated a root system. While a root system lives in the positive side
of the isotropy cone of its associated bilinear form, an imaginary cone lives
in the negative side of the isotropic cone. Precisely on the isotropic cone,
between root systems and imaginary cones, lives the set E of limit points of
the directions of roots (see arXiv:1112.5415). In this article we study the
close relations of the imaginary cone (see arXiv:1210.5206) with the set E,
which leads to new fundamental results about the structure of geometric
representations of infinite Coxeter groups. In particular, we show that the
W-action on E is minimal and faithful, and that E and the imaginary cone can be
approximated arbitrarily well by sets of limit roots and imaginary cones of
universal root subsystems of W, i.e., root systems for Coxeter groups without
braid relations (the free object for Coxeter groups). Finally, we discuss open
questions as well as the possible relevance of our framework in other areas
such as geometric group theory.Comment: v1: 63 pages, 14 figures. v2: Title changed; abstract and
introduction expanded and a few typos corrected. v3: 71 pages; some further
corrections after referee report, and many additions (most notably, relations
with geometric group theory (7.4) and Appendix on links with Benoist's limit
sets). To appear in Mathematische Zeitschrif
Garside families in Artin-Tits monoids and low elements in Coxeter groups
We show that every finitely generated Artin-Tits group admits a finite
Garside family, by introducing the notion of a low element in a Coxeter group
and proving that the family of all low elements in a Coxeter system (W, S) with
S finite includes S and is finite and closed under suffix and join with respect
to the right weak order
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