Bases: sage.combinat.root_system.ambient_space.AmbientSpace
EXAMPLES:
sage: RootSystem("A2xB2").ambient_space()
Ambient space of the Root system of type A2xB2
Returns a list of the irreducible Cartan types of which the given reducible Cartan type is a product.
EXAMPLES:
sage: RootSystem("A2xB2").ambient_space().ambient_spaces()
[Ambient space of the Root system of type ['A', 2],
Ambient space of the Root system of type ['B', 2]]
EXAMPLES:
sage: RootSystem("A2xB2").ambient_space().cartan_type()
A2xB2
EXAMPLES:
sage: RootSystem("A2xB2").ambient_space().component_types()
[['A', 2], ['B', 2]]
EXAMPLES:
sage: RootSystem("A2xB2").ambient_space().dimension()
5
EXAMPLES:
sage: RootSystem("A2xB2").ambient_space().fundamental_weights()
Finite family {1: (1, 0, 0, 0, 0), 2: (1, 1, 0, 0, 0), 3: (0, 0, 0, 1, 0), 4: (0, 0, 0, 1/2, 1/2)}
Produces the corresponding element of the lattice.
INPUT:
EXAMPLES:
sage: V = RootSystem("A2xB2").ambient_space()
sage: [V.inject_weights(i,V.ambient_spaces()[i].fundamental_weights()[1]) for i in range(2)]
[(1, 0, 0, 0, 0), (0, 0, 0, 1, 0)]
sage: [V.inject_weights(i,V.ambient_spaces()[i].fundamental_weights()[2]) for i in range(2)]
[(1, 1, 0, 0, 0), (0, 0, 0, 1/2, 1/2)]
EXAMPLES:
sage: RootSystem("A1xA2").ambient_space().negative_roots()
[(-1, 1, 0, 0, 0), (0, 0, -1, 1, 0), (0, 0, -1, 0, 1), (0, 0, 0, -1, 1)]
EXAMPLES:
sage: RootSystem("A1xA2").ambient_space().positive_roots()
[(1, -1, 0, 0, 0), (0, 0, 1, -1, 0), (0, 0, 1, 0, -1), (0, 0, 0, 1, -1)]
EXAMPLES:
sage: A = RootSystem(“A1xB2”).ambient_space() sage: A.simple_coroot(2) (0, 0, 1, -1) sage: A.simple_coroots() Finite family {1: (1, -1, 0, 0), 2: (0, 0, 1, -1), 3: (0, 0, 0, 2)}
EXAMPLES:
sage: A = RootSystem("A1xB2").ambient_space()
sage: A.simple_root(2)
(0, 0, 1, -1)
sage: A.simple_roots()
Finite family {1: (1, -1, 0, 0), 2: (0, 0, 1, -1), 3: (0, 0, 0, 1)}
Bases: sage.structure.sage_object.SageObject, sage.combinat.root_system.cartan_type.CartanType_abstract
A class for reducible Cartan types
alias of AmbientSpace
Returns an ascii art representation of this reducible Cartan type
EXAMPLES:
sage: print CartanType("F4xA2").ascii_art(label = lambda x: x+2)
O---O=>=O---O
3 4 5 6
O---O
7 8
sage: print CartanType(["BC",5,2], ["A",4]).ascii_art()
O=<=O---O---O---O=<=O
1 2 3 4 5 6
O---O---O---O
7 8 9 10
sage: print CartanType(["A",4], ["BC",5,2], ["C",3]).ascii_art()
O---O---O---O
1 2 3 4
O=<=O---O---O---O=<=O
5 6 7 8 9 10
O---O=<=O
11 12 13
Returns the Cartan matrix associated with self. By default the Cartan matrix is a subdivided block matrix showing the reducibility but the subdivision can be suppressed with the option subdivide=False.
EXAMPLES:
sage: ct = CartanType("A2","B2")
sage: ct.cartan_matrix()
[ 2 -1| 0 0]
[-1 2| 0 0]
[-----+-----]
[ 0 0| 2 -1]
[ 0 0|-2 2]
sage: ct.cartan_matrix(subdivide=False)
[ 2 -1 0 0]
[-1 2 0 0]
[ 0 0 2 -1]
[ 0 0 -2 2]
A list of Cartan types making up the reducible type.
EXAMPLES:
sage: CartanType(['A',2],['B',2]).component_types()
[['A', 2], ['B', 2]]
EXAMPLES:
sage: CartanType("A2xB2").dual()
A2xC2
Returns a Dynkin diagram for type reducible.
EXAMPLES:
sage: dd = CartanType("A2xB2xF4").dynkin_diagram()
sage: dd
O---O
1 2
O=>=O
3 4
O---O=>=O---O
5 6 7 8
A2xB2xF4
sage: dd.edges()
[(1, 2, 1), (2, 1, 1), (3, 4, 2), (4, 3, 1), (5, 6, 1), (6, 5, 1), (6, 7, 2), (7, 6, 1), (7, 8, 1), (8, 7, 1)]
sage: CartanType("F4xA2").dynkin_diagram()
O---O=>=O---O
1 2 3 4
O---O
5 6
F4xA2
Implements CartanType_abstract.index_set().
For the moment, the index set is always of the form .
EXAMPLES:
sage: CartanType("A2","A1").index_set()
[1, 2, 3]
Report that this reducible Cartan type is not affine
EXAMPLES:
sage: CartanType(['A',2],['B',2]).is_affine()
False
EXAMPLES:
sage: ct1 = CartanType(['A',2],['B',2])
sage: ct1.is_finite()
True
sage: ct2 = CartanType(['A',2],['B',2,1])
sage: ct2.is_finite()
False
TESTS:
sage: isinstance(ct1, sage.combinat.root_system.cartan_type.CartanType_finite)
True
sage: isinstance(ct2, sage.combinat.root_system.cartan_type.CartanType_finite)
False
Report that this Cartan type is not irreducible.
EXAMPLES:
sage: ct = CartanType(['A',2],['B',2])
sage: ct.is_irreducible()
False
Returns the rank of self.
EXAMPLES:
sage: CartanType("A2","A1").rank()
3
Returns “reducible” since the type is reducible.
EXAMPLES:
sage: CartanType(['A',2],['B',2]).type()
'reducible'