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# Root system data for type A

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

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

  EXAMPLES:

      sage: R = RootSystem(["A",3])
      sage: e = R.ambient_space(); e
      Ambient space of the Root system of type ['A', 3]
      sage: TestSuite(e).run()

  By default, this ambient space uses the barycentric projection for plotting:

      sage: L = RootSystem(["A",2]).ambient_space()
      sage: e = L.basis()
      sage: L._plot_projection(e[0])
      (1/2, 989/1142)
      sage: L._plot_projection(e[1])
      (-1, 0)
      sage: L._plot_projection(e[2])
      (1/2, -989/1142)
      sage: L = RootSystem(["A",3]).ambient_space()
      sage: l = L.an_element(); l
      (2, 2, 3, 0)
      sage: L._plot_projection(l)
      (0, -1121/1189, 7/3)

  See also

  - `sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods._plot_projection()`
  `det`(k=1)

    returns the vector (1, ... ,1) which in the [‘A’,r] weight lattice, interpreted as a weight of GL(r+1,CC) is the determinant. If the optional parameter k is given, returns (k, ... ,k), the k-th power of the determinant.

    EXAMPLES:

        sage: e = RootSystem(['A',3]).ambient_space()
        sage: e.det(1/2)
        (1/2, 1/2, 1/2, 1/2)

  `dimension`()

    EXAMPLES:

        sage: e = RootSystem(["A",3]).ambient_space()
        sage: e.dimension()
        4

  `fundamental_weight`(i)

    EXAMPLES:

        sage: e = RootSystem(['A',3]).ambient_lattice()
        sage: e.fundamental_weights()
        Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1, 1, 1, 0)}

  `highest_root`()
    EXAMPLE:
      sage: e = RootSystem([‘A’,3]).ambient_lattice() sage: e.highest_root() (1, 0, 0, -1)
  `negative_roots`()

    EXAMPLES:

        sage: e = RootSystem(['A',3]).ambient_lattice()
        sage: e.negative_roots()
        [(-1, 1, 0, 0),
         (-1, 0, 1, 0),
         (-1, 0, 0, 1),
         (0, -1, 1, 0),
         (0, -1, 0, 1),
         (0, 0, -1, 1)]

  `positive_roots`()

    EXAMPLES:

        sage: e = RootSystem(['A',3]).ambient_lattice()
        sage: e.positive_roots()
        [(1, -1, 0, 0),
         (1, 0, -1, 0),
         (0, 1, -1, 0),
         (1, 0, 0, -1),
         (0, 1, 0, -1),
         (0, 0, 1, -1)]

  `root`(i, j)

    Note that indexing starts at 0.

    EXAMPLES:

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

  `simple_root`(i)

    EXAMPLES:

        sage: e = RootSystem(['A',3]).ambient_lattice()
        sage: e.simple_roots()
        Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}

  classmethod `smallest_base_ring`(cartan_type=None)

    Returns the smallest base ring the ambient space can be defined upon

    See also

    `smallest_base_ring()`

    EXAMPLES:

        sage: e = RootSystem(["A",3]).ambient_space()
        sage: e.smallest_base_ring()
        Integer Ring

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

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

  Cartan Type \(A_n\)

  See also

  `CartanType()`

  `AmbientSpace`

    alias of `AmbientSpace`

  `ascii_art`(label=<function <lambda> at 0xef09d0d4>)

    Returns an ascii art representation of the Dynkin diagram

    EXAMPLES:

        sage: print CartanType(['A',0]).ascii_art()
        sage: print CartanType(['A',1]).ascii_art()
        O
        1
        sage: print CartanType(['A',3]).ascii_art()
        O---O---O
        1   2   3
        sage: print CartanType(['A',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(['A',4]).coxeter_number()
        5

  `dual_coxeter_number`()

    Return the dual Coxeter number associated with `self`.

    EXAMPLES:

        sage: CartanType(['A',4]).dual_coxeter_number()
        5

  `dynkin_diagram`()

    Returns the Dynkin diagram of type A.

    EXAMPLES:

        sage: a = CartanType(['A',3]).dynkin_diagram()
        sage: a
        O---O---O
        1   2   3
        A3
        sage: sorted(a.edges())
        [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 1)]

    TEST:

        sage: a = DynkinDiagram(['A',1])
        sage: a
        O
        1
        A1
        sage: a.vertices(), a.edges()
        ([1], [])

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