Abstract
Let $\Gamma$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $\Gamma$. Recall that, by a theorem of Hee and M\uhlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat \Gamma$. We denote by $\Phi^+$ the set of positive roots of $\Gamma$ and by $\hat \Phi^+$ the set of positive roots of $\hat \Gamma$. Let $E$ be a vector space over a field $\K$ having a basis in one-to-one correspondence with $\Phi^+$. The action of $G$ on $\Gamma$ induces an action of $G$ on $\Phi^+$, and therefore on $E$. We show that $E^G$ contains a linearly independent family of vectors naturally in one-to-one correspondence with $\hat \Phi^+$ and we determine exactly when this family is a basis of $E^G$. This question is motivated by the construction of Krammer's style linear representations for non simply laced Artin groups.
Highlights
They proved that some linear representation ψ : Bn → GL(E) of the braid group Bn previously introduced by Lawrence [22] is faithful
A useful information for us is that E is a vector space over the field K = Q(q, z) of rational functions in two variables q, z over Q, and has a natural basis of the form {ei, j | 1 ≤ i < j ≤ n}
For a finite laced triangle free Coxeter graph Γ, they constructed a linear representation ψ : AΓ → GL(E), they showed that this representation is always faithful on the Artin monoid A+Γ, and they showed that it is faithful on the whole group AΓ if Γ is of spherical type
Summary
Bigelow [1] and Krammer [20] proved that the braid groups are linear answering a historical question in the subject. For a finite laced triangle free Coxeter graph Γ, they constructed a linear representation ψ : AΓ → GL(E), they showed that this representation is always faithful on the Artin monoid A+Γ, and they showed that it is faithful on the whole group AΓ if Γ is of spherical type. An idea for constructing such linear representations for some Artin groups associated to non laced Coxeter graphs can be found in Digne [12]. Digne [12] proves that ψg is faithful and that Eg has a “natural” basis in one-to-one correspondence with the set Φ+ of positive roots of Γ This defines a linear representation for the Artin groups associated with the Coxeter graphs Bn (n ≥ 2), G2 and F4.
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