On the isomorphism problem, indecomposability and the automorphism groups of Coxeter groups

研究成果: Contribution to conferencePaper査読

抄録

Let (W, S) and (W′, S′) be Coxeter systems, {W λ} λ∈Λ and {W Λ ′′ } λ′∈Λ ′ the sets of the irreducible components of W relative to S and of W′ relative to S′ respectively, and let f : W → W′ be an isomorphism of abstract groups. Their Coxeter graphs may not be isomorphic. We show that f(Π λ∈Λ,|Wλ|<∞ W λ) = Π λ∈Λ,|Wλ′ ′|<∞W λ′′, and that there is a unique bijection φ: {λ ∈ Λ | |W λ| = ∞} → {λ ′ ∈ Λ ′ | |W λ′′| = ∞} such that f(W λ) ≡ W φ(λ)′ mod Z(W′) for every λ ∈ Λ with |W λ| = ∞, where Z(W′) is the center of W0. We also determine which two finite Coxeter groups are isomorphic. Our result reduces the problem of deciding whether two Coxeter groups are isomorphic to the case of infinite irreducible Coxeter groups. As a corollary we determine which irreducible Coxeter group is directly indecomposable as an abstract group. In particular, any infinite irreducible Coxeter group is directly indecomposable.

本文言語英語
ページ857-868
ページ数12
出版ステータス出版済み - 2005
外部発表はい
イベント17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05 - Taormina, イタリア
継続期間: 6 20 20056 25 2005

会議

会議17th Annual International Conference on Algebraic Combinatorics and Formal Power Series, FPSAC'05
国/地域イタリア
CityTaormina
Period6/20/056/25/05

All Science Journal Classification (ASJC) codes

  • 代数と数論

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