Weyl group

Conjugation of a Weyl element

Let Φ ⊂𝔼 be a root system, and 𝐺 be the group of all invertible linear endomorphisms leaving Φ invariant, i.e.

𝐺={𝜑∈GL⁡(𝔼):𝜑(Φ)=Φ}

Then the Weyl group W of Φ is a normal subgroup of 𝐺, in particular for any 𝛼 ∈Φ and 𝜏 ∈𝐺 geo

𝜏𝜎𝛼𝜏−1=𝜎𝜎(𝛼)

Therewithal¹

⟨𝛽,𝛼⟩=⟨𝜑(𝛽),𝜑(𝛼)⟩

Proof

proof

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Footnotes

1. 1972. Introduction to Lie Algebras and Representation Theory, §9.2, p. 43 ↩