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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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