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# Root system data for type B

class `sage.combinat.root_system.type_B.``AmbientSpace`(root_system, base_ring)

  Bases: `sage.combinat.root_system.ambient_space.AmbientSpace`

  EXAMPLES:

      sage: e = RootSystem(['A',3]).ambient_lattice()
      sage: s = e.simple_reflections()

      sage: L = RootSystem(['A',3]).coroot_lattice()
      sage: e.has_coerce_map_from(L)
      True
      sage: e(L.simple_root(1))
      (1, -1, 0, 0)

  `dimension`()

    EXAMPLES:

        sage: e = RootSystem(['B',3]).ambient_space()
        sage: e.dimension()
        3

  `fundamental_weight`(i)

    EXAMPLES:

        sage: RootSystem(['B',3]).ambient_space().fundamental_weights()
        Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1/2, 1/2, 1/2)}

  `negative_roots`()

    EXAMPLES:

        sage: RootSystem(['B',3]).ambient_space().negative_roots()
        [(-1, 1, 0),
         (-1, -1, 0),
         (-1, 0, 1),
         (-1, 0, -1),
         (0, -1, 1),
         (0, -1, -1),
         (-1, 0, 0),
         (0, -1, 0),
         (0, 0, -1)]

  `positive_roots`()

    EXAMPLES:

        sage: RootSystem(['B',3]).ambient_space().positive_roots()
        [(1, -1, 0),
         (1, 1, 0),
         (1, 0, -1),
         (1, 0, 1),
         (0, 1, -1),
         (0, 1, 1),
         (1, 0, 0),
         (0, 1, 0),
         (0, 0, 1)]

  `root`(i, j)

    Note that indexing starts at 0.

    EXAMPLES:

        sage: e = RootSystem(['B',3]).ambient_space()
        sage: e.root(0,1)
        (1, -1, 0)

  `simple_root`(i)

    EXAMPLES:

        sage: e = RootSystem(['B',4]).ambient_space()
        sage: e.simple_roots()
        Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1), 4: (0, 0, 0, 1)}
        sage: e.positive_roots()
        [(1, -1, 0, 0),
        (1, 1, 0, 0),
        (1, 0, -1, 0),
        (1, 0, 1, 0),
        (1, 0, 0, -1),
        (1, 0, 0, 1),
        (0, 1, -1, 0),
        (0, 1, 1, 0),
        (0, 1, 0, -1),
        (0, 1, 0, 1),
        (0, 0, 1, -1),
        (0, 0, 1, 1),
        (1, 0, 0, 0),
        (0, 1, 0, 0),
        (0, 0, 1, 0),
        (0, 0, 0, 1)]
        sage: e.fundamental_weights()
        Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1, 1, 1, 0), 4: (1/2, 1/2, 1/2, 1/2)}

class `sage.combinat.root_system.type_B.``CartanType`(n)

  Bases: `sage.combinat.root_system.cartan_type.CartanType_standard_finite`, `sage.combinat.root_system.cartan_type.CartanType_simple`, `sage.combinat.root_system.cartan_type.CartanType_crystallographic`

  EXAMPLES:

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

      sage: ct.is_irreducible()
      True
      sage: ct.is_finite()
      True
      sage: ct.is_affine()
      False
      sage: ct.is_crystallographic()
      True
      sage: ct.is_simply_laced()
      False
      sage: ct.affine()
      ['B', 4, 1]
      sage: ct.dual()
      ['C', 4]

      sage: ct = CartanType(['B',1])
      sage: ct.is_simply_laced()
      True
      sage: ct.affine()
      ['B', 1, 1]

  TESTS:

      sage: TestSuite(ct).run()

  `AmbientSpace`

    alias of `AmbientSpace`

  `ascii_art`(label=<function <lambda> at 0xef09d64c>)

    Return an ascii art representation of the Dynkin diagram.

    EXAMPLES:

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

  `coxeter_number`()

    Return the Coxeter number associated with `self`.

    EXAMPLES:

        sage: CartanType(['B',4]).coxeter_number()
        8

  `dual`()

    Types B and C are in duality:

    EXAMPLES:

        sage: CartanType(["C", 3]).dual()
        ['B', 3]

  `dual_coxeter_number`()

    Return the dual Coxeter number associated with `self`.

    EXAMPLES:

        sage: CartanType(['B',4]).dual_coxeter_number()
        7

  `dynkin_diagram`()

    Returns a Dynkin diagram for type B.

    EXAMPLES:

        sage: b = CartanType(['B',3]).dynkin_diagram()
        sage: b
        O---O=>=O
        1   2   3
        B3
        sage: sorted(b.edges())
        [(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)]

        sage: b = CartanType(['B',1]).dynkin_diagram()
        sage: b
        O
        1
        B1
        sage: sorted(b.edges())
        []

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