V

is a finite spanning set

Φ \subset V ∖ {0}

such that

Φ \cap R α = {α, - α} and s_{α} (Φ) = Φ (α \in Φ),

s_{α} (v) = v - 2 \frac{( v , α )}{( α , α )} α .

For a crystallographic root system one additionally requires

2 (β, α) / (α, α) \in Z

; that condition is not needed for general finite reflection groups.

Δ \subset Φ

such that every root has either all nonnegative or all nonpositive coordinates in this basis. Its associated positive system of a root system is

Π = Φ \cap span_{R_{\geq 0}} Δ,

and

Φ = Π ⊔ (- Π)

Solved by gpt-5.6-sol high.

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