Root systems - Combinatorics

Skip to content
Back to top
View this page
Edit this page

# 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

- Fusion 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