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

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 0x4425410>)

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
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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