Root Systems

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Quickref

• T = CartanType(["A", 3]), T.is_finite() – Cartan types

• T.dynkin_diagram(), DynkinDiagram(["G",2]) – Dynkin diagrams

• T.cartan_matrix(),  CartanMatrix(["F",4]) – Cartan matrices

• RootSystem(T).weight_lattice() – Root systems

• WeylGroup(["B", 6, 1]).simple_reflections() – Affine Weyl groups

• WeylCharacterRing(["D", 4]) – Weyl character rings

Introductory material

• Root Systems – This overview

• CartanType – An introduction to Cartan types

• RootSystem – An introduction to root systems

• Tutorial: visualizing root systems – A root system visualization tutorial

• The Lie Methods and Related Combinatorics thematic tutorial

Related material

• Crystals – Crystals

Cartan datum

• Cartan types

• Dynkin diagrams

• Cartan matrices

• Coxeter Matrices

• Coxeter Types

Root systems

• Root systems

• Tutorial: visualizing root systems

• Root lattice realizations

• Group algebras of root lattice realizations

• Weight lattice realizations

• Root lattices and root spaces

• Weight lattices and weight spaces

• Ambient lattices and ambient spaces

Coxeter groups

• Coxeter Groups

• Weyl Groups

• Extended Affine Weyl Groups

• Fundamental Group of an Extended Affine Weyl Group

• Braid Move Calculator

• Braid Orbit

See also

The categories CoxeterGroups and WeylGroups

Finite reflection groups

• Finite complex reflection groups

• Finite real reflection groups

See also

The category ComplexReflectionGroups

Representation theory

• Weyl Character Rings

• Fusion Rings

• F-Matrix Factory for FusionRings

• Integrable Representations of Affine Lie Algebras

• Branching Rules

• Hecke algebra representations

• Nonsymmetric Macdonald polynomials

Root system data and code for specific families of Cartan types

• Root system data for affine Cartan types

• Root system data for dual Cartan types

• Root system data for folded Cartan types

• Root system data for reducible Cartan types

• Root system data for relabelled Cartan types

• Root system data for Cartan types with marked nodes

Root system data and code for specific Cartan types

• Root system data for type A

• Root system data for type B

• Root system data for type C

• Root system data for type D

• Root system data for type E

• Root system data for type F

• Root system data for type G

• Root system data for type H

• Root system data for type I

• Root system data for (untwisted) type A affine

• Root system data for (untwisted) type B affine

• Root system data for (untwisted) type C affine

• Root system data for (untwisted) type D affine

• Root system data for (untwisted) type E affine

• Root system data for (untwisted) type F affine

• Root system data for (untwisted) type G affine

• Root system data for type BC affine

• Root system data for super type A

• Root system data for type A infinity