DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Weyl group

Definition

Let $Φ \subseteq E$ be a reduced crystallographic root system (Reduced crystallographic Euclidean root system) with coroots $α^{\lor} = 2 α / (α, α)$ (Coroot and dual root system). For $α \in Φ$ the associated reflection is $s_{α} (x) = x - (x, α^{\lor}) α = x - \frac{2 (x, α)}{(α, α)} α,$ which is an orthogonal transformation of $E$ with $s_{α} (α) = - α$ and $s_{α} (x) = x$ for $(x, α) = 0$ . The Weyl group of $Φ$ is the subgroup $W (Φ) = ⟨ s_{α} : α \in Φ ⟩ \leq O (E)$ of the orthogonal group generated by all root reflections.

Since $s_{α} (Φ) = Φ$ for every $α \in Φ$ by the root-system axioms, every element of $W (Φ)$ permutes $Φ$ . The Weyl group depends on $Φ$ and its inner product; rescaling the inner product does not change it, because the reflections $s_{α}$ are unchanged.

Depends on

Used by

Length and longest Weyl-group element Definition
Open and closed Weyl chambers Definition
The root system A₁ Example
The Weyl group of Aₙ is the symmetric group Example
Weyl groups of Bₙ and Dₙ Example
Simple reflections preserve weight multiplicities Lemma
Extremal Weyl-orbit weights Proposition
Highest weight of the dual representation Proposition
The Weyl group is finite and faithful Proposition
The Weyl vector in fundamental coordinates Proposition
Analytic and root-system Weyl groups agree Theorem
Rank-two root-system classification Theorem
Simple transitivity on Weyl chambers Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Chapter II (standard reference, not scraped)