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Weyl group
Definition
Let be a reduced crystallographic root system (Reduced crystallographic Euclidean root system) with coroots (Coroot and dual root system). For the associated reflection is which is an orthogonal transformation of with and for . The Weyl group of is the subgroup of the orthogonal group generated by all root reflections.
Since for every by the root-system axioms, every element of permutes . The Weyl group depends on and its inner product; rescaling the inner product does not change it, because the reflections are unchanged.
Depends on
Used by
- Abstract root data and their Weyl groups Definition
- Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group Definition
- Complex semisimple algebraic group, Borel, and flag variety Definition
- Length and longest Weyl-group element Definition
- Open and closed Weyl chambers Definition
- Standard parabolic and Levi subalgebras from a subset of simple roots Definition
- G2 from I2(6): the scaled realization and its twelve roots Example
- The A₁ affine line: alcoves, translations, and the root versus coroot lattice Example
- The extended affine Weyl group and non-trivial alcove stabilizers Example
- The invariants and the coinvariant Hilbert series of I₂(m): an explicit computation and the noncrystallographic contrast Example
- The root system A₁ Example
- The Weyl group of Aₙ is the symmetric group Example
- Weyl groups of Bₙ and Dₙ Example
- Affine reflections: translation form, involutivity, local finiteness, and Wₐ=Q^∨⋊ W Lemma
- Bruhat double cosets from rank-one multiplication Lemma
- Classical Poincare products for A, B, D and I2(m) from the permutation and signed-permutation models Lemma
- Combinatorics of a reduced root datum Lemma
- Crystallographic alcove diagrams: the affine list realized by Weyl types A–G Lemma
- Minimal parabolic from one negative simple root Lemma
- Point stabilizers, vertex residues, and rank-two boundary words Lemma
- Simple reflections preserve weight multiplicities Lemma
- Extremal Weyl-orbit weights Proposition
- Highest weight of the dual representation Proposition
- The Weyl group is finite and faithful Proposition
- The Weyl vector in fundamental coordinates Proposition
- Analytic and root-system Weyl groups agree Theorem
- Bruhat cells of the flag variety Theorem
- Crystallographic finite type: the Weyl types, reduced realizations and lattice stability Theorem
- Rank-two root-system classification Theorem
- Simple transitivity on Weyl chambers Theorem
Dependency tree · two levels
3 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)