Root reflections and the Weyl group action

Depends on

• The Killing-dual vector attached to a root
• The root set is a reduced crystallographic root system

Used by

• The center of the enveloping algebra is polynomial on rank-many generators Corollary
• The Racah--Speiser tensor-product algorithm Corollary
• Omitting the exterior root-weight shifts gives the wrong dot weight Counterexample
• Dot-Weyl facets and single-wall translation data Definition
• Minuscule weights Definition
• The classical BGG category O Definition
• The coinvariant algebra of a finite Weyl group Definition
• The integral Weyl group of a weight Definition
• The inversion set of a Weyl group element Definition
• The regular integral block and its antidominant weight Definition
• The strong linkage order on weights Definition
• The Weyl alternation operator Definition
• A singular integral A2 central-character summand Example
• An sl3 weight with cohomology in degree one Example
• Degree-one Kostant classes correspond to simple reflections Example
• Degree-two and degree-three Harish-Chandra generators for sl₃ Example
• Kostant cohomology for sl2 Example
• Kostant cohomology for the trivial sl3 module Example
• A dominant vector minimises its distance to a dominant weight Lemma
• A regular weight has a unique dominant dot translate Lemma
• Bruhat covers are right multiplication by positive-root reflections Lemma
• Dominant integral weights are maxima of their Weyl orbits Lemma
• Positive coroot pairings of a dominant integral weight Lemma
• Preliminary factorization of a Shapovalov determinant Lemma
• Rank-one cohomology shifts across a simple wall Lemma
• Singular dot weights have zero line-bundle cohomology Lemma
• The central reduction of U(sl2) is simple away from the finite-dimensional central characters Lemma
• The difference of the Weyl vector from its reflections is a sum of positive roots Lemma
• The extremal weight cochain of a Weyl element is closed and unique Lemma
• The rho-shift intertwines the dot and ordinary Weyl actions Lemma
• The shifted norm of a weight is maximal only at the top weight Lemma
• The sign of the Weyl length is multiplicative Lemma
• The simple-root singular vector in a Verma module Lemma
• The sum of basic Weyl invariant degrees minus one counts positive roots Lemma
• Weight subsets with equal root sums are unique Lemma
• Weyl alternants are skew-invariant 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
• Steinberg's tensor-product multiplicity formula Theorem

…and 1 more result.

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)