DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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Root reflections and the Weyl group action

Definition

Let $Φ$ be the root system from The root set is a reduced crystallographic root system. For a root $α$ , define the corresponding coroot by

$α^{\lor} : = \frac{2 H_{α}}{α (H_{α})} \in h,$

where $H_{α}$ is the vector from The Killing-dual vector attached to a root. The associated root reflection is the linear map

$s_{α} (λ) : = λ - λ (α^{\lor}) α (λ \in h^{*}) .$

The subgroup of $GL (h^{*})$ generated by these reflections is the Weyl group $W$ , acting on $h^{*}$ in the usual way.

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

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)