Root Systems

---

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
• 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