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

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

Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element

When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a 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 NCP_W as a generalised Fuss-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 NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map LL, describing the factorisations of its discriminant and its Jacobian. As byproducts, we generalise a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorisations of a Coxeter element of W.Comment: 18 pages. Version 2 : corrected typos and improved presentation. Version 3 : corrected typos, added illustrated example. To appear in Journal of Algebraic Combinatoric

Groupes de réflexion, géométrie du discriminant et partitions non-croisées

When W is a well-generated complex reflection group, the noncrossing partition lattice NCP_W of type W is a very rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. This structure appears in several algebraic setups (dual braid monoid, cluster algebras...). Many combinatorial properties of NCP_W are proved case-by-case, using the classification of reflection groups. It is the case for Chapoton's formula, expressing the number of multichains of a given length in the lattice NCP_W, in terms of the invariant degrees of W. This thesis work is motivated by the search for a geometric explanation of this formula, which could lead to a uniform understanding of the connections between the combinatorics of NCP_W and the invariant theory of W. The starting point is to use the Lyashko-Looijenga covering (LL), based on the geometry of the discriminant of W. In the first chapter, some topological constructions of Bessis are refined, allowing to relate the fibers of LL with block factorisations of a Coxeter element. Then we prove a transitivity property for the Hurwitz action of the braid group B_n on certain factorisations. Chapter 2 is devoted to certain finite polynomial extensions, and to properties about their Jacobians and discriminants. In Chapter 3, these results are applied to the extension defined by the covering LL. We deduce — with a case-free proof — formulas for the number of submaximal factorisations of a Coxeter element in W, in terms of the homogeneous degrees of the irreducible components of the discriminant and Jacobian for LL.Lorsque W est un groupe de réflexion complexe bien engendré, le treillis NCP_W des partitions non-croisées de type W est un objet combinatoire très riche, généralisant la notion de partitions non-croisées d'un n-gone, et intervenant dans divers contextes algébriques (monoïde de tresses dual, algèbres amassées...). De nombreuses propriétés combinatoires de NCP_W sont démontrées au cas par cas, à partir de la classification des groupes de réflexion. C'est le cas de la formule de Chapoton, qui exprime le nombre de chaînes de longueur donnée dans le treillis NCP_W en fonction des degrés invariants de W. Les travaux de cette thèse sont motivés par la recherche d'une explication géométrique de cette formule, qui permettrait une compréhension uniforme des liens entre la combinatoire de NCP_W et la théorie des invariants de W. Le point de départ est l'utilisation du revêtement de Lyashko-Looijenga (LL), défini à partir de la géométrie du discriminant de W. Dans le chapitre 1, on raffine des constructions topologiques de Bessis, permettant de relier les fibres de LL aux factorisations d'un élément de Coxeter. On établit ensuite une propriété de transitivité de l'action d'Hurwitz du groupe de tresses B_n sur certaines factorisations. Le chapitre 2 porte sur certaines extensions finies d'anneaux de polynômes, et sur des propriétés concernant leurs jacobiens et leurs discriminants. Dans le chapitre 3, on applique ces résultats au cas des extensions définies par un revêtement LL. On en déduit — sans utiliser la classification — des formules donnant le nombre de factorisations sous-maximales d'un élément de Coxeter de W en fonction des degrés homogènes des composantes irréductibles du discriminant et du jacobien de LL