On the Limit Set of Root Systems of Coxeter Groups acting on Lorentzian spaces

Hohlweg, Christophe; Préaux, Jean-Philippe; Ripoll, Vivien

doi:10.48550/arxiv.1305.0052

Cited by 5 publications

(11 citation statements)

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“…Another motivation is to study discrete subgroups of isometries in quadratic spaces; for instance modules associated to geometric representations of W are quadratic spaces and W is itself a discrete subgroup of isometries generated by reflections. The case of Lorentzian spaces is discussed in [HPR13] but the results here suggest such a study may be of considerable interest more generally.…”

Section: Introductionmentioning

confidence: 92%

“…9]. These relations are explored in the general context of Lorentzian spaces in the article [HPR13], and are outlined in §7.4 at the end of this article. 4.3.…”

Section: Is Clear That the Domainmentioning

confidence: 99%

“…Some of our results are generalizations of known theorems in Kleinian group theory, such as the minimality of the action. In [HPR13], which was written after a first version of this paper was circulated, the authors explore some of the relations between hyperbolic geometry and our setting. If Φ is weakly hyperbolic, it means that Φ is a root system in the Lorentzian space (V, B), which contains models for the hyperbolic space H n , where n + 1 = dim(V ).…”

Section: Existence Of Reflection Subgroups That Are Universal Coxeter...mentioning

confidence: 99%

“…In particular, in §7.4, we discuss the relations of our framework with hyperbolic geometry and geometric group theory. The link with Kleinian groups, which is precisely discussed in another article [HPR13], is outlined, and we also explain for instance how the convex core of W is related to the imaginary convex set and the limit roots. Considering that our framework and results apply more generally to discrete subgroups of isometries on quadratic spaces, an important question we raise…”

Section: Introductionmentioning

confidence: 99%

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Imaginary cones and limit roots of infinite Coxeter groups

2016

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Abstract. 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 isotropic 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. In this article we study the close relations of the imaginary cone 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.

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Section: Introductionmentioning

confidence: 92%

“…9]. These relations are explored in the general context of Lorentzian spaces in the article [HPR13], and are outlined in §7.4 at the end of this article. 4.3.…”

Section: Is Clear That the Domainmentioning

confidence: 99%

Section: Existence Of Reflection Subgroups That Are Universal Coxeter...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Imaginary cones and limit roots of infinite Coxeter groups

2016

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“…In particular, one should try to have a better understanding of the real roots: they form a nice subset of the set of all positive roots and the representation theory attached to these roots is better understood. In [12,13,14], the authors have studied accumulation points of real roots in more general root systems, where accumulation has a meaning when one looks at the rays associated to the roots. In a root system defined from an acyclic quiver, it is straightforward to check that the real roots always accumulate to the hypersurface defined by the Tits form of the quiver.…”

Section: Introductionmentioning

confidence: 99%

Accumulation points of real Schur roots

Paquette

2016

Journal of Algebra

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Let k be an algebraically closed field and Q be an acyclic quiver with n vertices. Consider the category rep(Q) of finite dimensional representations of Q over k. The exceptional representations of Q, that is, the indecomposable objects of rep(Q) without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to Q. These roots are special elements of the Grothendieck group Z n of rep(Q). When we identify the dimension vectors of the representations (that is, the non-negative vectors of Z n ) up to positive multiple, we see that the real Schur roots can accumulate in some directions of R n ⊃ Z n . This paper is devoted to the study of these accumulation points. After giving new properties of the canonical decomposition of dimension vectors, we show how to use this decomposition to describe the rational accumulation points. Finally, we study the irrational accumulation points and we give a complete description of them in case Q is of weakly hyperbolic type.

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Lorentzian Coxeter systems and Boyd–Maxwell ball packings

Chen

Labbé

2014

Geom Dedicata

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In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: Given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd-Maxwell ball packings. Finally, we enumerate all the Coxeter systems that generate Boyd-Maxwell ball packings.2010 Mathematics Subject Classification. Primary 52C17, 20F55; Secondary 05C30.

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On the Limit Set of Root Systems of Coxeter Groups acting on Lorentzian spaces

Cited by 5 publications

References 0 publications

Imaginary cones and limit roots of infinite Coxeter groups

Imaginary cones and limit roots of infinite Coxeter groups

Accumulation points of real Schur roots

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

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