Normalizers of parabolic subgroups in Coxeter groups

Brink, Brigitte; Howlett, Robert B.

doi:10.1007/s002220050312

Cited by 70 publications

(167 citation statements)

References 8 publications

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“…As mentioned in Section 1.1, a similar problem of describing the normalizer N W ðW I Þ of a parabolic subgroup W I was studied for finite W by Howlett [14] and for general W by Brink and Howlett [5]. Some part of our result is closely related to their result, however some part of our result is essentially new and not derived from their work.…”

Section: Related Workmentioning

confidence: 68%

“…The following theorem, used in [5] for general S, is proven in [8] by Deodhar under the assumption jSj < y. A proof for the general case is given in [21].…”

Section: The Groupoid Cmentioning

confidence: 99%

“…Section 2 is devoted to some preliminaries, and Section 3 summarizes the properties of the groupoid C that are obtained by lifting the corresponding properties of the groupoid G. Section 4 investigates the groupoid C further, enhancing its similarity with Coxeter groups noticed by Brink and Howlett in [5], and derive the decomposition of the vertex group C I ¼ W ?I z Y I as well as the presentations of W ?I and Y I . (Although this looks similar to the decomposition G I ¼Ñ N I z M I given in [5], there seems no overall relation between the structures ofÑ N I and W ?I or between those of M I and Y I ; see Remark 4.7 for details.) Section 5 completes the description of the entire Z W ðW I Þ, and gives a description of N W ðW I Þ based on the same argument.…”

Section: Related Workmentioning

confidence: 99%

“…In this section, we introduce and study a groupoid C, one of whose vertex groups gives a fairly large part (though not all) of the centralizer Z W ðW I Þ. As we shall see in Section 3.3, our groupoid C is a covering of the groupoid G defined by Brink and Howlett in [5] for their study of the normalizer N W ðW I Þ. As a consequence, many properties, but not all, of C can be obtained from those of G.…”

Section: The Groupoid Cmentioning

confidence: 99%

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On centralizers of parabolic subgroups in Coxeter groups

Nuida

2011

JGTH

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Abstract. Let W be an arbitrary Coxeter group, possibly of infinite rank. We describe a decomposition of the centralizer Z W ðW I Þ of an arbitrary parabolic subgroup W I into the center of W I , a Coxeter group and a subgroup defined by a 2-cell complex. Only information about finite parabolic subgroups is required in an explicit computation. By using our description of Z W ðW I Þ, we will be able to reveal a further strong property of the action of the third factor on the second factor, in particular on the finite irreducible components of the second factor.

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Section: Related Workmentioning

confidence: 68%

“…The following theorem, used in [5] for general S, is proven in [8] by Deodhar under the assumption jSj < y. A proof for the general case is given in [21].…”

Section: The Groupoid Cmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: The Groupoid Cmentioning

confidence: 99%

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On centralizers of parabolic subgroups in Coxeter groups

Nuida

2011

JGTH

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“…If (W, S) is even, and T ⊂ S is such that T is finite, then T decomposes as a direct product of groups each factor which is dihedral or Z 2 . It is straightforward to see that none of the finite triangle groups (2,3,3), (2,3,4) and (2,3,5) are isomorphic to a subgroup of a direct product of dihedral groups. Tits' result then implies that these finite triangle groups are not subgroups of an even Coxeter group.…”

Section: A Matching Theorem and The Proof Of Propositionmentioning

confidence: 99%

The even isomorphism theorem for Coxeter groups

Mihalik

2007

Trans. Amer. Math. Soc.

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Abstract. Coxeter groups have presentations S : (st) m st ∀s, t ∈ S where for all s, t ∈ S, m st ∈ {1, 2, . . . , ∞}, m st = m ts and m st = 1 if and only if s = t. A fundamental question in the theory of Coxeter groups is: Given two such "Coxeter" presentations, do they present the same group? There are two known ways to change a Coxeter presentation, generally referred to as twisting and simplex exchange. We solve the isomorphism question for Coxeter groups with an even Coxeter presentation (one in which m st is even or ∞ when s = t). More specifically, we give an algorithm that describes a sequence of twists and triangle-edge exchanges that either converts an arbitrary finitely generated Coxeter presentation into a unique even presentation or identifies the group as a non-even Coxeter group. Our technique can be used to produce all Coxeter presentations for a given even Coxeter group.

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Automorphisms of Coxeter Groups of Rank 3 with Infinite Bonds

Franzsen

2002

Journal of Algebra

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If W is a rank 3 Coxeter group, whose Coxeter diagram contains at least one infinite bond, then the automorphism group of W is larger than the group generated by the inner automorphisms and the automorphisms induced from automorphisms of the Coxeter diagram. In each case the automorphism group of W is described. PRELIMINARIESThe aim of this paper is to describe the automorphism group of rank 3 Coxeter groups whose diagram contains at least one infinite bond.We recall that a Coxeter group is a group with a presentation of the form W = gp r a a ∈ r a r b m ab = 1 for all a b ∈ where is some indexing set, whose cardinality is called the rank of W , and the parameters m ab satisfy the following conditions: m ab = m ba , each m ab lies in the set m ∈ m ≥ 1 ∪ ∞ , and m ab = 1 if and only if a = b. To simplify matters write r i for the generator r a i . If w ∈ W then we define l w to be the length of the shortest expression for w as a product of generators r a (a ∈ ).The (Coxeter) diagram of W is a graph with vertex set in which an edge (or bond) labeled m ab joins a b ∈ whenever m ab ≥ 3. We say that the group is irreducible if this graph is connected.Let V be a real vector space with basis , and define a bilinear form B on V by B a b = − cos π/m ab for all a b ∈ with m ab < ∞ and B a b = −1 otherwise. For each a ∈ V such that B a a = 1 we define

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Normalizers of parabolic subgroups in Coxeter groups

Cited by 70 publications

References 8 publications

On centralizers of parabolic subgroups in Coxeter groups

On centralizers of parabolic subgroups in Coxeter groups

The even isomorphism theorem for Coxeter groups

Automorphisms of Coxeter Groups of Rank 3 with Infinite Bonds

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