DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The integral Weyl group of a weight

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For a weight $λ$ , define

$Φ_{λ} = {α \in Φ : ⟨ λ + ρ, α^{\lor} ⟩ \in Z}, W_{λ} = ⟨ s_{α} : α \in Φ_{λ} ⟩ \subset W .$

Reflections and coroots are those of Root reflections and the Weyl group action, with the shift from The Weyl vector rho for a chosen positive system. The integral-reflection linkage class through $λ$ is $W_{λ} \cdot λ$ . The word integral includes zero and negative integral pairings. If $Φ_{λ}$ is empty the generated group is ${1}$ . This definition uses generating reflections; no identification with a root-lattice-coset stabilizer is assumed.

Depends on

Used by

Dependency tree · two levels

5 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

Humphreys, §3.4 definitions and invariance identities, p.52; §4.9, p.83 (reflection-generated convention) (standard reference, not scraped)