Root system data for relabelled Cartan types

class sage.combinat.root_system.type_relabel.AmbientSpace(root_system, base_ring)

Bases: sage.combinat.root_system.ambient_space.AmbientSpace

Ambient space for a relabelled finite Cartan type.

It is constructed in the canonical way from the ambient space of the original Cartan type, by relabelling the simple roots, fundamental weights, etc.

EXAMPLES:

sage: cycle = {1:2, 2:3, 3:4, 4:1}
sage: L = CartanType(["F",4]).relabel(cycle).root_system().ambient_space(); L
Ambient space of the Root system of type ['F', 4] relabelled by {1: 2, 2: 3, 3: 4, 4: 1}
sage: TestSuite(L).run()
dimension()

Return the dimension of this ambient space.

EXAMPLES:

sage: cycle = {1:2, 2:3, 3:4, 4:1}
sage: L = CartanType(["F",4]).relabel(cycle).root_system().ambient_space()
sage: L.dimension()
4
fundamental_weight(i)

Return the i-th fundamental weight.

It is constructed by looking up the corresponding simple coroot in the ambient space for the original Cartan type.

EXAMPLES:

sage: cycle = {1:2, 2:3, 3:4, 4:1}
sage: L = CartanType(["F",4]).relabel(cycle).root_system().ambient_space()
sage: K = CartanType(["F",4]).root_system().ambient_space()
sage: K.fundamental_weights()
Finite family {1: (1, 1, 0, 0), 2: (2, 1, 1, 0), 3: (3/2, 1/2, 1/2, 1/2), 4: (1, 0, 0, 0)}
sage: L.fundamental_weight(1)
(1, 0, 0, 0)
sage: L.fundamental_weights()
Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (2, 1, 1, 0), 4: (3/2, 1/2, 1/2, 1/2)}
simple_root(i)

Return the i-th simple root.

It is constructed by looking up the corresponding simple coroot in the ambient space for the original Cartan type.

EXAMPLES:

sage: cycle = {1:2, 2:3, 3:4, 4:1}
sage: L = CartanType(["F",4]).relabel(cycle).root_system().ambient_space()
sage: K = CartanType(["F",4]).root_system().ambient_space()
sage: K.simple_roots()
Finite family {1: (0, 1, -1, 0), 2: (0, 0, 1, -1), 3: (0, 0, 0, 1), 4: (1/2, -1/2, -1/2, -1/2)}
sage: K.simple_coroots()
Finite family {1: (0, 1, -1, 0), 2: (0, 0, 1, -1), 3: (0, 0, 0, 2), 4: (1, -1, -1, -1)}
sage: L.simple_root(1)
(1/2, -1/2, -1/2, -1/2)

sage: L.simple_roots()
Finite family {1: (1/2, -1/2, -1/2, -1/2), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1), 4: (0, 0, 0, 1)}

sage: L.simple_coroots()
Finite family {1: (1, -1, -1, -1), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1), 4: (0, 0, 0, 2)}
class sage.combinat.root_system.type_relabel.CartanType(type, relabelling)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.sage_object.SageObject, sage.combinat.root_system.cartan_type.CartanType_abstract

A class for relabelled Cartan types.

ascii_art(label=<function <lambda> at 0x5b039b0>)

Returns an ascii art representation of this Cartan type

EXAMPLES:

sage: print CartanType(["G", 2]).relabel({1:2,2:1}).ascii_art()
  3
O=<=O
2   1
sage: print CartanType(["B", 3, 1]).relabel([1,3,2,0]).ascii_art()
    O 1
    |
    |
O---O=>=O
3   2   0
sage: print CartanType(["F", 4, 1]).relabel(lambda n: 4-n).ascii_art()
O---O---O=>=O---O
4   3   2   1   0
dual()

Implements sage.combinat.root_system.cartan_type.CartanType_abstract.dual(), using that taking the dual and relabelling are commuting operations.

EXAMPLES:

sage: T = CartanType(["BC",3, 2])
sage: cycle = {1:2, 2:3, 3:0, 0:1}
sage: T.relabel(cycle).dual().dynkin_diagram()
O=>=O---O=>=O
1   2   3   0
BC3~* relabelled by {0: 1, 1: 2, 2: 3, 3: 0}
sage: T.dual().relabel(cycle).dynkin_diagram()
O=>=O---O=>=O
1   2   3   0
BC3~* relabelled by {0: 1, 1: 2, 2: 3, 3: 0}
dynkin_diagram()

Returns the dynkin diagram for this Cartan type.

EXAMPLES:

sage: CartanType(["G", 2]).relabel({1:2,2:1}).dynkin_diagram()
  3
O=<=O
2   1
G2 relabelled by {1: 2, 2: 1}

TESTS:

To be compared with the examples in ascii_art():

sage: sorted(CartanType(["G", 2]).relabel({1:2,2:1}).dynkin_diagram().edges())
[(1, 2, 3), (2, 1, 1)]
sage: sorted(CartanType(["B", 3, 1]).relabel([1,3,2,0]).dynkin_diagram().edges())
[(0, 2, 1), (1, 2, 1), (2, 0, 2), (2, 1, 1), (2, 3, 1), (3, 2, 1)]
sage: sorted(CartanType(["F", 4, 1]).relabel(lambda n: 4-n).dynkin_diagram().edges())
[(0, 1, 1), (1, 0, 1), (1, 2, 1), (2, 1, 2), (2, 3, 1), (3, 2, 1), (3, 4, 1), (4, 3, 1)]
index_set()

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.index_set()
(1, 2)
is_affine()

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.is_affine()
False
is_crystallographic()

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.is_crystallographic()
True
is_finite()

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.is_finite()
True
is_irreducible()

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.is_irreducible()
True
rank()

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.rank()
2
type()

Return the type of self or None if unknown.

EXAMPLES:

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})
sage: ct.type()
'G'
class sage.combinat.root_system.type_relabel.CartanType_affine(type, relabelling)

Bases: sage.combinat.root_system.type_relabel.CartanType, sage.combinat.root_system.cartan_type.CartanType_affine

TESTS:

sage: ct = CartanType(['D',4,3]); ct
['G', 2, 1]^* relabelled by {0: 0, 1: 2, 2: 1}

sage: L = ct.root_system().ambient_space(); L
Ambient space of the Root system of type ['G', 2, 1]^* relabelled by {0: 0, 1: 2, 2: 1}
sage: L.classical()
Ambient space of the Root system of type ['G', 2]
sage: TestSuite(L).run()
classical()

Return the classical Cartan type associated with self.

EXAMPLES:

sage: A41 = CartanType(['A',4,1])
sage: A41.dynkin_diagram()
0
O-----------+
|           |
|           |
O---O---O---O
1   2   3   4
A4~

sage: T = A41.relabel({0:1, 1:2, 2:3, 3:4, 4:0})
sage: T
['A', 4, 1] relabelled by {0: 1, 1: 2, 2: 3, 3: 4, 4: 0}
sage: T.dynkin_diagram()
1
O-----------+
|           |
|           |
O---O---O---O
2   3   4   0
A4~ relabelled by {0: 1, 1: 2, 2: 3, 3: 4, 4: 0}

sage: T0 = T.classical()
sage: T0
['A', 4] relabelled by {1: 2, 2: 3, 3: 4, 4: 0}
sage: T0.dynkin_diagram()
O---O---O---O
2   3   4   0
A4 relabelled by {1: 2, 2: 3, 3: 4, 4: 0}
is_untwisted_affine()

Implements :meth:’CartanType_affine.is_untwisted_affine`

A relabelled Cartan type is untwisted affine if the original is.

EXAMPLES:

sage: CartanType(['B', 3, 1]).relabel({1:2, 2:3, 3:0, 0:1}).is_untwisted_affine()
True
special_node()

Returns a special node of the Dynkin diagram

See also

special_node()

It is obtained by relabelling of the special node of the non relabelled Dynkin diagram.

EXAMPLES:

sage: CartanType(['B', 3, 1]).special_node()
0
sage: CartanType(['B', 3, 1]).relabel({1:2, 2:3, 3:0, 0:1}).special_node()
1
class sage.combinat.root_system.type_relabel.CartanType_finite(type, relabelling)

Bases: sage.combinat.root_system.type_relabel.CartanType, sage.combinat.root_system.cartan_type.CartanType_finite

INPUT:

  • type – a Cartan type
  • relabelling – a function (or a list, or a dictionary)

Returns an isomorphic Cartan type obtained by relabelling the nodes of the dynkin diagram. Namely the node with label i is relabelled f(i) (or, by f[i] if f is a list or dictionary).

EXAMPLES:

We take the Cartan type \(B_4\):

sage: T = CartanType(['B',4])
sage: T.dynkin_diagram()
O---O---O=>=O
1   2   3   4
B4

And relabel its nodes:

sage: cycle = {1:2, 2:3, 3:4, 4:1}

sage: T = T.relabel(cycle)
sage: T.dynkin_diagram()
O---O---O=>=O
2   3   4   1
B4 relabelled by {1: 2, 2: 3, 3: 4, 4: 1}
sage: sorted(T.dynkin_diagram().edges())
[(1, 4, 1), (2, 3, 1), (3, 2, 1), (3, 4, 1), (4, 1, 2), (4, 3, 1)]

Multiple relabelling are recomposed into a single one:

sage: T = T.relabel(cycle)
sage: T.dynkin_diagram()
O---O---O=>=O
3   4   1   2
B4 relabelled by {1: 3, 2: 4, 3: 1, 4: 2}

sage: T = T.relabel(cycle)
sage: T.dynkin_diagram()
O---O---O=>=O
4   1   2   3
B4 relabelled by {1: 4, 2: 1, 3: 2, 4: 3}

And trivial relabelling are honoured nicely:

sage: T = T.relabel(cycle)
sage: T.dynkin_diagram()
O---O---O=>=O
1   2   3   4
B4

Test that the produced cartan type is in the appropriate abstract classes (see trac ticket #13724):

sage: ct = CartanType(['B',4]).relabel(cycle)
sage: TestSuite(ct).run()
sage: from sage.combinat.root_system import cartan_type
sage: isinstance(ct, cartan_type.CartanType_finite)
True
sage: isinstance(ct, cartan_type.CartanType_simple)
True
sage: isinstance(ct, cartan_type.CartanType_affine)
False
sage: isinstance(ct, cartan_type.CartanType_crystallographic)
True
sage: isinstance(ct, cartan_type.CartanType_simply_laced)
False

sage: ct = CartanType(['A',3,1]).relabel({0:3,1:2, 2:1,3:0})
sage: TestSuite(ct).run()
sage: isinstance(ct, cartan_type.CartanType_simple)
True
sage: isinstance(ct, cartan_type.CartanType_finite)
False
sage: isinstance(ct, cartan_type.CartanType_affine)
True
sage: isinstance(ct, cartan_type.CartanType_crystallographic)
True
sage: isinstance(ct, cartan_type.CartanType_simply_laced)
True

Check for the original issues of trac ticket #13724:

sage: A3 = CartanType("A3")
sage: A3.cartan_matrix()
[ 2 -1  0]
[-1  2 -1]
[ 0 -1  2]
sage: A3r = A3.relabel({1:2,2:3,3:1})
sage: A3r.cartan_matrix()
[ 2  0 -1]
[ 0  2 -1]
[-1 -1  2]

sage: ct = CartanType(["D",4,3]).classical(); ct
['G', 2]
sage: ct.symmetrizer()
Finite family {1: 1, 2: 3}
AmbientSpace

alias of AmbientSpace

affine()

Return the affine Cartan type associated with self.

EXAMPLES:

sage: B4 = CartanType(['B',4])
sage: B4.dynkin_diagram()
O---O---O=>=O
1   2   3   4
B4
sage: B4.affine().dynkin_diagram()
    O 0
    |
    |
O---O---O=>=O
1   2   3   4
B4~

If possible, this reuses the original label for the special node:

sage: T = B4.relabel({1:2, 2:3, 3:4, 4:1}); T.dynkin_diagram()
O---O---O=>=O
2   3   4   1
B4 relabelled by {1: 2, 2: 3, 3: 4, 4: 1}
sage: T.affine().dynkin_diagram()
    O 0
    |
    |
O---O---O=>=O
2   3   4   1
B4~ relabelled by {0: 0, 1: 2, 2: 3, 3: 4, 4: 1}

Otherwise, it chooses a label for the special_node in \(0,1,...\):

sage: T = B4.relabel({1:0, 2:1, 3:2, 4:3}); T.dynkin_diagram()
O---O---O=>=O
0   1   2   3
B4 relabelled by {1: 0, 2: 1, 3: 2, 4: 3}
sage: T.affine().dynkin_diagram()
    O 4
    |
    |
O---O---O=>=O
0   1   2   3
B4~ relabelled by {0: 4, 1: 0, 2: 1, 3: 2, 4: 3}

This failed before trac ticket #13724:

sage: ct = CartanType(["G",2]).dual(); ct
['G', 2] relabelled by {1: 2, 2: 1}
sage: ct.affine()
['G', 2, 1] relabelled by {0: 0, 1: 2, 2: 1}

sage: ct = CartanType(["F",4]).dual(); ct
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
sage: ct.affine()
['F', 4, 1] relabelled by {0: 0, 1: 4, 2: 3, 3: 2, 4: 1}

TESTS:

Check that we don’t inadvertently change the internal relabelling of ct:

sage: ct
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}

Previous topic

Root system data for reducible Cartan types

Next topic

Root system data for type A

This Page