Root system data for (untwisted) type G affine

class sage.combinat.root_system.type_G_affine.CartanType

Bases: CartanType_standard_untwisted_affine

EXAMPLES:

Sage

sage: ct = CartanType(['G',2,1])
sage: ct
['G', 2, 1]
sage: ct._repr_(compact = True)
'G2~'

sage: ct.is_irreducible()
True
sage: ct.is_finite()
False
sage: ct.is_affine()
True
sage: ct.is_untwisted_affine()
True
sage: ct.is_crystallographic()
True
sage: ct.is_simply_laced()
False
sage: ct.classical()
['G', 2]
sage: ct.dual()
['G', 2, 1]^*
sage: ct.dual().is_untwisted_affine()
False

Python

>>> from sage.all import *
>>> ct = CartanType(['G',Integer(2),Integer(1)])
>>> ct
['G', 2, 1]
>>> ct._repr_(compact = True)
'G2~'

>>> ct.is_irreducible()
True
>>> ct.is_finite()
False
>>> ct.is_affine()
True
>>> ct.is_untwisted_affine()
True
>>> ct.is_crystallographic()
True
>>> ct.is_simply_laced()
False
>>> ct.classical()
['G', 2]
>>> ct.dual()
['G', 2, 1]^*
>>> ct.dual().is_untwisted_affine()
False

ascii_art(label=None, node=None)

Returns an ascii art representation of the Dynkin diagram

EXAMPLES:

Sage

sage: print(CartanType(['G',2,1]).ascii_art(label = lambda x: x+2))
  3
O=<=O---O
3   4   2

Python

>>> from sage.all import *
>>> print(CartanType(['G',Integer(2),Integer(1)]).ascii_art(label = lambda x: x+Integer(2)))
  3
O=<=O---O
3   4   2

dynkin_diagram()

Returns the extended Dynkin diagram for type G.

EXAMPLES:

Sage

sage: g = CartanType(['G',2,1]).dynkin_diagram(); g                         # needs sage.graphs
  3
O=<=O---O
1   2   0
G2~
sage: g.edges(sort=True)                                                    # needs sage.graphs
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 3)]

Python

>>> from sage.all import *
>>> g = CartanType(['G',Integer(2),Integer(1)]).dynkin_diagram(); g                         # needs sage.graphs
  3
O=<=O---O
1   2   0
G2~
>>> g.edges(sort=True)                                                    # needs sage.graphs
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 3)]