Root system data for (untwisted) type D affine - Combinatorics

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# 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()

    Return 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)]