Root system data for (untwisted) type B affine

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

Bases: CartanType_standard_untwisted_affine

EXAMPLES:

Sage

sage: ct = CartanType(['B',4,1])
sage: ct
['B', 4, 1]
sage: ct._repr_(compact = True)
'B4~'

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()
['B', 4]
sage: ct.dual()
['B', 4, 1]^*
sage: ct.dual().is_untwisted_affine()
False

Python

>>> from sage.all import *
>>> ct = CartanType(['B',Integer(4),Integer(1)])
>>> ct
['B', 4, 1]
>>> ct._repr_(compact = True)
'B4~'

>>> 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()
['B', 4]
>>> ct.dual()
['B', 4, 1]^*
>>> ct.dual().is_untwisted_affine()
False

PieriFactors

alias of PieriFactors_type_B_affine

ascii_art(label=None, node=None)

Return an ascii art representation of the extended Dynkin diagram.

EXAMPLES:

Sage

sage: print(CartanType(['B',3,1]).ascii_art())
    O 0
    |
    |
O---O=>=O
1   2   3

sage: print(CartanType(['B',5,1]).ascii_art(label = lambda x: x+2))
    O 2
    |
    |
O---O---O---O=>=O
3   4   5   6   7

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

Python

>>> from sage.all import *
>>> print(CartanType(['B',Integer(3),Integer(1)]).ascii_art())
    O 0
    |
    |
O---O=>=O
1   2   3

>>> print(CartanType(['B',Integer(5),Integer(1)]).ascii_art(label = lambda x: x+Integer(2)))
    O 2
    |
    |
O---O---O---O=>=O
3   4   5   6   7

>>> print(CartanType(['B',Integer(2),Integer(1)]).ascii_art(label = lambda x: x+Integer(2)))
O=>=O=<=O
2   4   3
>>> print(CartanType(['B',Integer(1),Integer(1)]).ascii_art(label = lambda x: x+Integer(2)))
O<=>O
2   3

dynkin_diagram()

Return the extended Dynkin diagram for affine type \(B\).

EXAMPLES:

Sage

sage: # needs sage.graphs
sage: b = CartanType(['B',3,1]).dynkin_diagram(); b
    O 0
    |
    |
O---O=>=O
1   2   3
B3~
sage: b.edges(sort=True)
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]
sage: b = CartanType(['B',2,1]).dynkin_diagram(); b
O=>=O=<=O
0   2   1
B2~
sage: b.edges(sort=True)
[(0, 2, 2), (1, 2, 2), (2, 0, 1), (2, 1, 1)]
sage: b = CartanType(['B',1,1]).dynkin_diagram(); b
O<=>O
0   1
B1~
sage: b.edges(sort=True)
[(0, 1, 2), (1, 0, 2)]

Python

>>> from sage.all import *
>>> # needs sage.graphs
>>> b = CartanType(['B',Integer(3),Integer(1)]).dynkin_diagram(); b
    O 0
    |
    |
O---O=>=O
1   2   3
B3~
>>> b.edges(sort=True)
[(0, 2, 1), (1, 2, 1), (2, 0, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]
>>> b = CartanType(['B',Integer(2),Integer(1)]).dynkin_diagram(); b
O=>=O=<=O
0   2   1
B2~
>>> b.edges(sort=True)
[(0, 2, 2), (1, 2, 2), (2, 0, 1), (2, 1, 1)]
>>> b = CartanType(['B',Integer(1),Integer(1)]).dynkin_diagram(); b
O<=>O
0   1
B1~
>>> b.edges(sort=True)
[(0, 1, 2), (1, 0, 2)]