Alphabeta MathHow statement and proof provenance work
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Fundamental weights for a chosen simple root system
Definition
Fix a simple root system inside the root system of The root set is a reduced crystallographic root system. The simple coroots form a basis of , so there are unique functionals satisfying
These functionals are the fundamental weights for the chosen simple root system.
Depends on
Used by
- Changing the line-bundle sign changes the Borel-Weil section space Counterexample
- An sl3 weight with cohomology in degree one Example
- Borel-Weil-Bott on the projective line for SL2 Example
- Flag line bundles for SL(2) Example
- The sl2 singular weight has no cohomology Example
- The top-degree Borel-Weil-Bott case and Serre duality Example
- Two minimal-parabolic projections for SL(3) Example
- Highest- and lowest-weight submodules at the exceptional parameters Lemma
- Weyl-invariant polynomials on the Cartan extend to invariant polynomials on g Lemma
- Finite-dimensional simple modules are classified by dominant highest weights Theorem
- Generic irreducibility and the exceptional parameter lattice Theorem
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)