# Algorithm to study if a vector within the geometric illustration of a Coxeter group is proportional to a root retort

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## Algorithm to study if a vector within the geometric illustration of a Coxeter group is proportional to a root

Let $$W$$ breathe a Coxeter group, and let $$V$$ breathe its geometric illustration (as outlined for example in Section 5.3 of Humphrey’s bespeak Reflection teams and Coxeter teams). Let $$vin Vbackslash{0}$$ (within the illustration by which I’m one might occupy that $$v$$ is a $$mathbb{Q}$$-linear mixture of constructive roots).

Is there an algorithm permitting one to study in finite time if $$v$$ is proportional to a root ?

If $$W$$ is finite that is limpid (on can write down the matrix of the reflection sending $$v$$ to $$-v$$ and preserving the Tits design, and examine with the matrices of reflections in $$W$$), however I’m within the illustration the place $$W$$ is innumerable.

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