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Coroot and dual root system
Definition
Let be a reduced crystallographic root system in the real inner product space (Reduced crystallographic Euclidean root system). For put The vector is the coroot of , and the set is the dual root system.
The definitions are well posed because for . Two elementary identities, obtained by substituting the definition and using bilinearity of the inner product, are for all ; in particular , and the Cartan integer equals the inner product . Also is a positive real multiple of , so and .
Depends on
Used by
- Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group Definition
- Cartan matrix of a based root system Definition
- Crystallographic scalings: scaled simple roots, coroots and the root, coroot and weight lattices Definition
- Root, coroot, weight, and coweight lattices Definition
- Weyl group Definition
- Flag line bundles for SL(2) Example
- Root versus coroot translation lattices: A2, B2 and two conventions Example
- Simple roots and fundamental weights of Aₙ Example
- The A₁ affine line: alcoves, translations, and the root versus coroot lattice Example
- The A₂ and B₂ fundamental alcoves: coordinates, corner vectors, and facet types Example
- The A₂ root and weight lattices: P/Q has order three Example
- The extended affine Weyl group and non-trivial alcove stabilizers Example
- The two realizations of I₂(4): B₂ and C₂ with their lattices and duality Example
- Two minimal-parabolic projections for SL(3) Example
- Affine reflections: translation form, involutivity, local finiteness, and Wₐ=Q^∨⋊ W Lemma
- Chevalley basis and real structure constants Lemma
- Complex-weight separation of Harish-Chandra characters Lemma
- Crystallographic alcove diagrams: the affine list realized by Weyl types A–G Lemma
- Flag line-bundle degree on a minimal-parabolic fiber Lemma
- Minuscule weights have exactly the Weyl orbit as their weights Lemma
- Point stabilizers, vertex residues, and rank-two boundary words Lemma
- Rank-one SL2 homomorphism and Weyl representative Lemma
- Duality exchanges B and C Proposition
- The Weyl group is finite and faithful Proposition
- The Weyl vector in fundamental coordinates Proposition
- Crystallographic finite type: the Weyl types, reduced realizations and lattice stability Theorem
- Rank-two root-system classification Theorem
- The Jantzen sum formula for a Verma module Theorem
Dependency tree · two levels
2 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)