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

How statement and proof provenance work

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Fundamental weights for a chosen simple root system

Definition

Fix a simple root system $Δ = {α_{1}, \dots, α_{r}}$ inside the root system of The root set is a reduced crystallographic root system. The simple coroots $α_{i}^{\lor}$ form a basis of $h$ , so there are unique functionals $ω_{1}, \dots, ω_{r} \in h^{*}$ satisfying

$ω_{i} (α_{j}^{\lor}) = δ_{i j} (1 \leq i, j \leq r) .$

These functionals are the fundamental weights for the chosen simple root system.

Depends on

The root set is a reduced crystallographic root system

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)