On the rank of a Coxeter group

Ratcliffe, John G.

doi:10.1515/jgt.2008.089

Cited by 3 publications

(7 citation statements)

References 3 publications

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“…The proof of this is split into two parts, one of which is published in , the other covering the major part of the present paper. The paper uses only basic results from the theory of Coxeter systems and elementary (but less known) techniques from . This is not true for the second part of the proof of Theorem 1, presented here, because it uses the full power of the main result in .…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…The question of reversing the blowing up construction for Coxeter generating sets was addressed by Mihalik and Ratcliffe in . They introduced a blowing down procedure, which decreases the rank.…”

Section: Spherical Principal Components and Blowing Downmentioning

confidence: 99%

“…Here we adopt this latter terminology and call the elements s ∈ S satisfying the conditions above blowing up generators for (W, S). The question of reversing the blowing up construction for Coxeter generating sets was addressed by Mihalik and Ratcliffe in [17]. They introduced a blowing down procedure, which decreases the rank.…”

Section: Spherical Principal Components and Blowing Downmentioning

confidence: 99%

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Intrinsic reflections and strongly rigid Coxeter groups

Howlett

Mühlherr

Nuida

2017

Proc. London Math. Soc.

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It is possible for a group W that is abstractly isomorphic to a Coxeter group to have more than one conjugacy class of Coxeter generating sets, and if S and R are two non‐conjugate Coxeter generating sets then it may or may not be the case that some element s∈S is conjugate to an element r∈R. In this paper we classify the so‐called intrinsic reflections: those elements of W whose conjugacy class intersects non‐trivially every Coxeter generating set. In combination with previously known results, this leads us to a classification of Coxeter groups for which all Coxeter generating sets are conjugate.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

Section: Spherical Principal Components and Blowing Downmentioning

confidence: 99%

Section: Spherical Principal Components and Blowing Downmentioning

confidence: 99%

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Intrinsic reflections and strongly rigid Coxeter groups

Howlett

Mühlherr

Nuida

2017

Proc. London Math. Soc.

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“…Indeed, the case of right-angled generators is the only one where the blowing down procedure for Coxeter generating sets comes into play. The latter has been introduced in [12] and some of our intermediate results should be compared with those in [12]. However, our treatment is completely independent because our assumptions and goals are quite different from those in [12].…”

Section: Introductionmentioning

confidence: 99%

“…The latter has been introduced in [12] and some of our intermediate results should be compared with those in [12]. However, our treatment is completely independent because our assumptions and goals are quite different from those in [12].…”

Section: Introductionmentioning

confidence: 99%