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""" 

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 

--------------------- 

 

- :ref:`sage.combinat.root_system` -- This overview 

- :class:`CartanType` -- An introduction to Cartan types 

- :class:`RootSystem` -- An introduction to root systems 

- :ref:`sage.combinat.root_system.plot` -- A root system visualization tutorial 

 

- The `Lie Methods and Related Combinatorics <../../../../../thematic_tutorials/lie.html>`_ thematic tutorial 

 

 

Related material 

---------------- 

 

- :ref:`sage.combinat.crystals` -- Crystals 

 

Cartan datum 

------------ 

 

- :ref:`sage.combinat.root_system.cartan_type` 

- :ref:`sage.combinat.root_system.dynkin_diagram` 

- :ref:`sage.combinat.root_system.cartan_matrix` 

- :ref:`sage.combinat.root_system.coxeter_matrix` 

- :ref:`sage.combinat.root_system.coxeter_type` 

 

Root systems 

------------ 

 

- :ref:`sage.combinat.root_system.root_system` 

- :ref:`sage.combinat.root_system.plot` 

- :ref:`sage.combinat.root_system.root_lattice_realizations` 

- :ref:`sage.combinat.root_system.root_lattice_realization_algebras` 

- :ref:`sage.combinat.root_system.weight_lattice_realizations` 

- :ref:`sage.combinat.root_system.root_space` 

- :ref:`sage.combinat.root_system.weight_space` 

- :ref:`sage.combinat.root_system.ambient_space` 

 

Coxeter groups 

-------------- 

 

- :ref:`sage.combinat.root_system.coxeter_group` 

- :ref:`sage.combinat.root_system.weyl_group` 

- :ref:`sage.combinat.root_system.extended_affine_weyl_group` 

- :ref:`sage.combinat.root_system.fundamental_group` 

- :ref:`sage.combinat.root_system.braid_move_calculator` 

 

.. SEEALSO:: 

 

The categories :class:`CoxeterGroups` and :class:`WeylGroups` 

 

Finite reflection groups 

------------------------ 

 

- :ref:`sage.combinat.root_system.reflection_group_complex` 

- :ref:`sage.combinat.root_system.reflection_group_real` 

 

.. SEEALSO:: 

 

The category :class:`~sage.categories.complex_reflection_groups.ComplexReflectionGroups` 

 

Representation theory 

--------------------- 

 

- :ref:`sage.combinat.root_system.weyl_characters` 

- :ref:`sage.combinat.root_system.integrable_representations` 

- :ref:`sage.combinat.root_system.branching_rules` 

- :ref:`sage.combinat.root_system.hecke_algebra_representation` 

- :ref:`sage.combinat.root_system.non_symmetric_macdonald_polynomials` 

 

Root system data and code for specific families of Cartan types 

--------------------------------------------------------------- 

 

- :ref:`sage.combinat.root_system.type_affine` 

- :ref:`sage.combinat.root_system.type_dual` 

- :ref:`sage.combinat.root_system.type_folded` 

- :ref:`sage.combinat.root_system.type_reducible` 

- :ref:`sage.combinat.root_system.type_relabel` 

- :ref:`sage.combinat.root_system.type_marked` 

 

Root system data and code for specific Cartan types 

--------------------------------------------------- 

 

- :ref:`sage.combinat.root_system.type_A` 

- :ref:`sage.combinat.root_system.type_B` 

- :ref:`sage.combinat.root_system.type_C` 

- :ref:`sage.combinat.root_system.type_D` 

- :ref:`sage.combinat.root_system.type_E` 

- :ref:`sage.combinat.root_system.type_F` 

- :ref:`sage.combinat.root_system.type_G` 

- :ref:`sage.combinat.root_system.type_H` 

- :ref:`sage.combinat.root_system.type_I` 

- :ref:`sage.combinat.root_system.type_A_affine` 

- :ref:`sage.combinat.root_system.type_B_affine` 

- :ref:`sage.combinat.root_system.type_C_affine` 

- :ref:`sage.combinat.root_system.type_D_affine` 

- :ref:`sage.combinat.root_system.type_E_affine` 

- :ref:`sage.combinat.root_system.type_F_affine` 

- :ref:`sage.combinat.root_system.type_G_affine` 

- :ref:`sage.combinat.root_system.type_BC_affine` 

- :ref:`sage.combinat.root_system.type_super_A` 

- :ref:`sage.combinat.root_system.type_A_infinity` 

""" 

# currently needed to activate the backward compatibility 

# register_unpickle_override 

from . import type_A 

from . import type_B 

from . import type_C 

from . import type_D 

from . import type_E 

from . import type_F 

from . import type_G