Root system data for (untwisted) type D affine

class sage.combinat.root_system.type_D_affine.CartanType(n)

Bases: CartanType_standard_untwisted_affine, CartanType_simply_laced

EXAMPLES:

Sage

sage: ct = CartanType(['D',4,1])
sage: ct
['D', 4, 1]
sage: ct._repr_(compact = True)
'D4~'

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()
True
sage: ct.classical()
['D', 4]
sage: ct.dual()
['D', 4, 1]

Python

>>> from sage.all import *
>>> ct = CartanType(['D',Integer(4),Integer(1)])
>>> ct
['D', 4, 1]
>>> ct._repr_(compact = True)
'D4~'

>>> 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()
True
>>> ct.classical()
['D', 4]
>>> ct.dual()
['D', 4, 1]

PieriFactors

alias of PieriFactors_type_D_affine

ascii_art(label=None, node=None)

Return an ascii art representation of the extended Dynkin diagram.

dynkin_diagram()

Returns the extended Dynkin diagram for affine type D.

EXAMPLES:

Sage

sage: d = CartanType(['D', 6, 1]).dynkin_diagram(); d                        # needs sage.graphs
   0 O       O 6
     |       |
     |       |
 O---O---O---O---O
 1   2   3   4   5
 D6~
sage: d.edges(sort=True)                                                     # needs sage.graphs
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 1),
 (3, 2, 1), (3, 4, 1), (4, 3, 1), (4, 5, 1), (4, 6, 1), (5, 4, 1), (6, 4, 1)]

sage: d = CartanType(['D', 4, 1]).dynkin_diagram(); d                        # needs sage.graphs
    O 4
    |
    |
O---O---O
1   |2  3
    |
    O 0
D4~
sage: d.edges(sort=True)                                                     # needs sage.graphs
[(0, 2, 1),
 (1, 2, 1),
 (2, 0, 1),
 (2, 1, 1),
 (2, 3, 1),
 (2, 4, 1),
 (3, 2, 1),
 (4, 2, 1)]

sage: d = CartanType(['D', 3, 1]).dynkin_diagram(); d                        # needs sage.graphs
0
O-------+
|       |
|       |
O---O---O
3   1   2
D3~
sage: d.edges(sort=True)                                                     # needs sage.graphs
[(0, 2, 1), (0, 3, 1), (1, 2, 1), (1, 3, 1),
 (2, 0, 1), (2, 1, 1), (3, 0, 1), (3, 1, 1)]

Python

>>> from sage.all import *
>>> d = CartanType(['D', Integer(6), Integer(1)]).dynkin_diagram(); d                        # needs sage.graphs
   0 O       O 6
     |       |
     |       |
 O---O---O---O---O
 1   2   3   4   5
 D6~
>>> d.edges(sort=True)                                                     # needs sage.graphs
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 1),
 (3, 2, 1), (3, 4, 1), (4, 3, 1), (4, 5, 1), (4, 6, 1), (5, 4, 1), (6, 4, 1)]

>>> d = CartanType(['D', Integer(4), Integer(1)]).dynkin_diagram(); d                        # needs sage.graphs
    O 4
    |
    |
O---O---O
1   |2  3
    |
    O 0
D4~
>>> d.edges(sort=True)                                                     # needs sage.graphs
[(0, 2, 1),
 (1, 2, 1),
 (2, 0, 1),
 (2, 1, 1),
 (2, 3, 1),
 (2, 4, 1),
 (3, 2, 1),
 (4, 2, 1)]

>>> d = CartanType(['D', Integer(3), Integer(1)]).dynkin_diagram(); d                        # needs sage.graphs
0
O-------+
|       |
|       |
O---O---O
3   1   2
D3~
>>> d.edges(sort=True)                                                     # needs sage.graphs
[(0, 2, 1), (0, 3, 1), (1, 2, 1), (1, 3, 1),
 (2, 0, 1), (2, 1, 1), (3, 0, 1), (3, 1, 1)]