Reflection subgroups of Coxeter systems

Dyer, Matthew

doi:10.1016/0021-8693(90)90149-i

Cited by 174 publications

(215 citation statements)

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“…The classification (b) was conjectured (in equivalent form) and partly proved by Coxeter [5] and was proved in [8]. The proof here via the stronger result (a) is along similar lines, but it is much simpler and independent of the classification of affine Weyl groups.…”

Section: Subsystems Of Affine Root Systems Imentioning

confidence: 77%

“…A classification (up to isomorphism) of the reflection subgroups of an affine Weyl group in terms of those of the corresponding finite Weyl group was conjectured and partly proved by Coxeter [5], and completed in [8]. The paper [12] includes explicit lists of the possible isomorphism types of maximal reflection subgroups and describes a procedure by which all reflection subgroups of a given affine Weyl group can be obtained.…”

Section: The Plan Of This Workmentioning

confidence: 99%

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Reflection subgroups of finite and affine Weyl groups

Dyer

Lehrer

2011

Trans. Amer. Math. Soc.

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Abstract. We give complete classifications of the reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case-free proof is given of the well-known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where we give two distinct 'on the nose' classifications of reflection subgroups, as well as reproving Coxeter's conjecture on the isomorphism classes of reflection groups which occur. Geometric and combinatorial aspects of root systems play an essential role. Parameter sets for various types of subsets of roots are interpreted in terms of alcove geometry and the Tits cone, and combinatorial identities are derived.

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Section: Subsystems Of Affine Root Systems Imentioning

confidence: 77%

Section: The Plan Of This Workmentioning

confidence: 99%

Reflection subgroups of finite and affine Weyl groups

Dyer

Lehrer

2011

Trans. Amer. Math. Soc.

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“…If W ⊂ W is a reflection subgroup, it is a well-known theorem due (independently) to Deodhar [3] and Dyer [4] that W is a Coxeter group w.r.t. a set S = {s γ i : i = 1 · · · k} of reflection generators.…”

Section: 2mentioning

confidence: 99%

“…These are subgroups W ⊂ W generated by reflections (conjugates of elements of S); they exhibit many of the same properties as parabolic subgroups [3], [4], e.g. (i) W is a Coxeter group in its own right, (ii) ∃ unique minimal length left coset representatives for W in W , etc.…”

Section: Introductionmentioning

confidence: 99%

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Untitled

2008

Proc. Amer. Math. Soc.

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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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Reflection subgroups of Coxeter systems

Cited by 174 publications

References 2 publications

Reflection subgroups of finite and affine Weyl groups

Reflection subgroups of finite and affine Weyl groups

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Automorphisms of Coxeter Groups of Rank 3 with Infinite Bonds

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