Positive-root criterion for Coxeter length

In the geometric representation of a Coxeter system,

ℓ (w s_{i}) > ℓ (w) ⟺ w (e_{i}) is a positive root,

and

ℓ (s_{i} w) > ℓ (w) ⟺ w^{- 1} (e_{i}) is a positive root .

Replacing “positive” by “negative” reverses either length inequality.

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