Imaginary cones and limit roots of infinite Coxeter groups

Dyer, Matthew S.; Hohlweg, Christophe; Ripoll, Vivien

doi:10.48550/arxiv.1303.6710

Cited by 7 publications

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“…In a subsequent paper [DHR13], we show that this property holds nevertheless for the set E 2 studied in §4, i.e., E 2 (Φ I ) = E 2 (Φ) ∩ F I . We also define other natural smaller dense subsets of E for which this property of parabolic restriction works.…”

Section: Now Letmentioning

confidence: 67%

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Asymptotical Behaviour of Roots of Infinite Coxeter Groups

2014

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Abstract. Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone Q 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 how 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.

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Section: Now Letmentioning

confidence: 67%

“…In the case of an hyperbolic cut, only one branch intersects conv(π H (∆)). In a following paper [DHR13], we prove and use the fact that for a hyperbolic group of rank 3, there always exists a hyperplane H, transverse to Φ + , such that Q ∩ H is a circle (the analogous property is actually valid for hyperbolic groups of higher rank).…”

Section: 2mentioning

confidence: 99%

Asymptotical Behaviour of Roots of Infinite Coxeter Groups

2014

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“…In many examples of Lorentzian Coxeter systems, fractal patterns of ball packings appear while visualizing limit roots on an affine hyperplane; see [HLR14, Figure 1(b)], [HPR13, Figure 1] and Figure 1 of the present article. A description of this fractal structure is conjectured in [HLR14, Section 3.2] and proved in [DHR13,Theorem 4.10]. In [HPR13], Hohlweg, Préaux and Ripoll prove that the set of limit roots of a Coxeter group W acting on a Lorentz space is equal to the limit set of W seen as a discrete subgroup of hyperbolic isometries.…”

Section: Introductionmentioning

confidence: 96%

“…The cone over limit roots is the imaginary cone [Dye13]. The relations between limit roots and the imaginary cone are further investigated in [DHR13].…”

Section: Introductionmentioning

confidence: 99%

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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“…When W is finite, there is a wellknown correspondence between reflection orders and reduced words for the longest element. When W is infinite, the collections of biclosed sets and reflections orders are not completely understood; see [10] or [11] for conjectures and recent progress. If W is a Weyl group, a slightly different definition of convex order is used.…”

Section: Introductionmentioning

confidence: 99%

Biclosed sets in real hyperplane arrangements

McConville¹

2014

Preprint

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The set of chambers of a real hyperplane arrangement may be ordered by separation from some fixed chamber. When this poset is a lattice, Björner, Edelman, and Ziegler proved that the chambers are in natural bijection with the biconvex sets of the arrangement. Two families of examples of arrangements with a lattice of chambers are simplicial and supersolvable arrangements. For these arrangements, we prove the that the chambers correspond to biclosed sets, a weakening of the biconvex property.

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Imaginary cones and limit roots of infinite Coxeter groups

Cited by 7 publications

References 0 publications

Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Asymptotical Behaviour of Roots of Infinite Coxeter Groups

Lorentzian Coxeter systems and Boyd–Maxwell ball packings

Biclosed sets in real hyperplane arrangements

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