20 research outputs found

### Connectivity Properties of Factorization Posets in Generated Groups

We consider three notions of connectivity and their interactions in partially
ordered sets coming from reduced factorizations of an element in a generated
group. While one form of connectivity essentially reflects the connectivity of
the poset diagram, the other two are a bit more involved: Hurwitz-connectivity
has its origins in algebraic geometry, and shellability in topology. We propose
a framework to study these connectivity properties in a uniform way. Our main
tool is a certain linear order of the generators that is compatible with the
chosen element.Comment: 35 pages, 17 figures. Comments are very welcome. Final versio

### Orbites d'Hurwitz des factorisations primitives d'un \'el\'ement de Coxeter

We study the Hurwitz action of the classical braid group on factorisations of
a Coxeter element c in a well-generated complex reflection group W. It is
well-known that the Hurwitz action is transitive on the set of reduced
decompositions of c in reflections. Our main result is a similar property for
the primitive factorisations of c, i.e. factorisations with only one factor
which is not a reflection. The motivation is the search for a geometric proof
of Chapoton's formula for the number of chains of given length in the
non-crossing partitions lattice NCP_W. Our proof uses the properties of the
Lyashko-Looijenga covering and the geometry of the discriminant of W.Comment: 25 pages, in French (Abstract in English). Version 3 : last version,
published in Journal of Algebra (typos corrected, some minor changes

### 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

### Submaximal factorizations of a Coxeter element in complex reflection groups

When $W$ is a finite reflection group, the noncrossing partition lattice $NC(W)$ of type $W$ is a very rich combinatorial object, extending the notion of noncrossing partitions of an $n$-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $NC(W)$ as a generalized FuĂź-Catalan number, depending on the invariant degrees of $W$. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $NC(W)$ as fibers of a "Lyashko-Looijenga covering''. This covering is constructed from the geometry of the discriminant hypersurface of $W$. We deduce new enumeration formulas for certain factorizations of a Coxeter element of $W$

### Asymptotical behaviour of roots of infinite Coxeter groups

Let W be an infinite Coxeter group. We initiate the study of the set E of
limit points of "normalized" positive roots (representing the directions of the
roots) of W. We show that E is contained in the isotropic cone of the bilinear
form B associated to a geometric representation, and illustrate this property
with numerous examples and pictures in rank 3 and 4. We also define a natural
geometric action of W on E, and then we exhibit a countable subset of E, formed
by limit points for the dihedral reflection subgroups of W. We explain that
this subset is built from the intersection with Q of the lines passing through
two positive roots, and finally we establish that it is dense in E.Comment: 19 pages, 11 figures. Version 2: 29 pages, 11 figures. Reorganisation
of the paper, addition of many details (section 5 in particular). Version 3 :
revised edition accepted in Journal of the CMS. The number "I" was removed
from the title since number "II" paper was named differently, see
http://arxiv.org/abs/1303.671

### On non-conjugate Coxeter elements in well-generated reflection groups

Given an irreducible well-generated complex reflection group $W$ with Coxeter number $h$, we call a Coxeter element any regular element (in the sense of Springer) of order $h$ in $W$; this is a slight extension of the most common notion of Coxeter element. We show that the class of these Coxeter elements forms a single orbit in $W$ under the action of reflection automorphisms. For Coxeter and Shephard groups, this implies that an element $c$ is a Coxeter element if and only if there exists a simple system $S$ of reflections such that $c$ is the product of the generators in $S$. We moreover deduce multiple further implications of this property. In particular, we obtain that all noncrossing partition lattices of $W$ associated to different Coxeter elements are isomorphic. We also prove that there is a simply transitive action of the Galois group of the field of definition of $W$ on the set of conjugacy classes of Coxeter elements. Finally, we extend several of these properties to Springer's regular elements of arbitrary order

### Asymptotical behaviour of roots of infinite Coxeter groups I

Let $W$ be an infinite Coxeter group, and $\Phi$ be the root system constructed from its geometric representation. We study the set $E$ of limit points of "normalized'' roots (representing the directions of the roots). We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form associated to $W$, and illustrate this property with numerous examples and pictures in rank $3$ and $4$. We also define a natural geometric action of $W$ on $E$, for which $E$ is stable. Then we exhibit a countable subset $E_2$ of $E$, formed by limit points for the dihedral reflection subgroups of $W$; we explain how $E_2$ can be built from the intersection with $Q$ of the lines passing through two roots, and we establish that $E_2$ is dense in $E$