Coxeter groups, symmetries, and rooted representations

2016/11/28 by Olivier Geneste, Luis Paris, Geneste, Olivier +1
Mathematics · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics

paper · doi:10.48550/arxiv.1611.09150

Abstract

Let (W,S) be a Coxeter system, let G be a group of symmetries of (W,S) and let f : W → \GL (V) be the linear representation associated with a root basis (V, ⟨ .,. ⟩, Π).We assume that G ⊂ \GL (V), and that G leaves invariant Π and ⟨ .,. ⟩. We show that W^G is a Coxeter group, we construct a subset Π⊂ V^G so that (V^G, ⟨ .,. ⟩, Π) is a root basis of W^G, and we show that the induced representation f^G : W^G → \GL(V^G) is the linear representation associated with (V^G, ⟨ .,. ⟩, Π).In particular, the latter is faithful. The fact that W^G is a Coxeter group is already known and is due to Mühlherr and Hée, but also follows directly from the proof of the other results.

Related

• Root systems, symmetries and linear representations of Artin groups
• A Construction of Coxeter Group Representations (II)
• On paired root systems of Coxeter groups
• Coxeter Elements and Root Bases
• A Unified Construction of Coxeter Group Representations (I)