Root system data for (untwisted) type E affine

class sage.combinat.root_system.type_E_affine.CartanType(n)[source]

Bases: CartanType_standard_untwisted_affine, CartanType_simply_laced

EXAMPLES:

sage: ct = CartanType(['E',6,1])
sage: ct
['E', 6, 1]
sage: ct._repr_(compact = True)
'E6~'

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()
['E', 6]
sage: ct.dual()
['E', 6, 1]

>>> from sage.all import *
>>> ct = CartanType(['E',Integer(6),Integer(1)])
>>> ct
['E', 6, 1]
>>> ct._repr_(compact = True)
'E6~'

>>> 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()
['E', 6]
>>> ct.dual()
['E', 6, 1]
ascii_art(label=None, node=None)[source]

Return an ascii art representation of the extended Dynkin diagram.

EXAMPLES:

sage: print(CartanType(['E',6,1]).ascii_art(label = lambda x: x+2))
        O 2
        |
        |
        O 4
        |
        |
O---O---O---O---O
3   5   6   7   8
sage: print(CartanType(['E',7,1]).ascii_art(label = lambda x: x+2))
            O 4
            |
            |
O---O---O---O---O---O---O
2   3   5   6   7   8   9
sage: print(CartanType(['E',8,1]).ascii_art(label = lambda x: x-3))
        O -1
        |
        |
O---O---O---O---O---O---O---O
-2  0   1   2   3   4   5   -3

>>> from sage.all import *
>>> print(CartanType(['E',Integer(6),Integer(1)]).ascii_art(label = lambda x: x+Integer(2)))
        O 2
        |
        |
        O 4
        |
        |
O---O---O---O---O
3   5   6   7   8
>>> print(CartanType(['E',Integer(7),Integer(1)]).ascii_art(label = lambda x: x+Integer(2)))
            O 4
            |
            |
O---O---O---O---O---O---O
2   3   5   6   7   8   9
>>> print(CartanType(['E',Integer(8),Integer(1)]).ascii_art(label = lambda x: x-Integer(3)))
        O -1
        |
        |
O---O---O---O---O---O---O---O
-2  0   1   2   3   4   5   -3
dynkin_diagram()[source]

Return the extended Dynkin diagram for affine type E.

EXAMPLES:

sage: e = CartanType(['E', 6, 1]).dynkin_diagram(); e                       # needs sage.graphs
        O 0
        |
        |
        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6
E6~
sage: e.edges(sort=True)                                                    # needs sage.graphs
[(0, 2, 1),
 (1, 3, 1),
 (2, 0, 1),
 (2, 4, 1),
 (3, 1, 1),
 (3, 4, 1),
 (4, 2, 1),
 (4, 3, 1),
 (4, 5, 1),
 (5, 4, 1),
 (5, 6, 1),
 (6, 5, 1)]

sage: # needs sage.graphs
sage: e = CartanType(['E', 7, 1]).dynkin_diagram(); e
            O 2
            |
            |
O---O---O---O---O---O---O
0   1   3   4   5   6   7
E7~
sage: e.edges(sort=True)
[(0, 1, 1), (1, 0, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
 (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
 (6, 5, 1), (6, 7, 1), (7, 6, 1)]
sage: e = CartanType(['E', 8, 1]).dynkin_diagram(); e
        O 2
        |
        |
O---O---O---O---O---O---O---O
1   3   4   5   6   7   8   0
E8~
sage: e.edges(sort=True)
[(0, 8, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
 (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
 (6, 5, 1), (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 0, 1), (8, 7, 1)]

>>> from sage.all import *
>>> e = CartanType(['E', Integer(6), Integer(1)]).dynkin_diagram(); e                       # needs sage.graphs
        O 0
        |
        |
        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6
E6~
>>> e.edges(sort=True)                                                    # needs sage.graphs
[(0, 2, 1),
 (1, 3, 1),
 (2, 0, 1),
 (2, 4, 1),
 (3, 1, 1),
 (3, 4, 1),
 (4, 2, 1),
 (4, 3, 1),
 (4, 5, 1),
 (5, 4, 1),
 (5, 6, 1),
 (6, 5, 1)]

>>> # needs sage.graphs
>>> e = CartanType(['E', Integer(7), Integer(1)]).dynkin_diagram(); e
            O 2
            |
            |
O---O---O---O---O---O---O
0   1   3   4   5   6   7
E7~
>>> e.edges(sort=True)
[(0, 1, 1), (1, 0, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
 (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
 (6, 5, 1), (6, 7, 1), (7, 6, 1)]
>>> e = CartanType(['E', Integer(8), Integer(1)]).dynkin_diagram(); e
        O 2
        |
        |
O---O---O---O---O---O---O---O
1   3   4   5   6   7   8   0
E8~
>>> e.edges(sort=True)
[(0, 8, 1), (1, 3, 1), (2, 4, 1), (3, 1, 1), (3, 4, 1),
 (4, 2, 1), (4, 3, 1), (4, 5, 1), (5, 4, 1), (5, 6, 1),
 (6, 5, 1), (6, 7, 1), (7, 6, 1), (7, 8, 1), (8, 0, 1), (8, 7, 1)]