Root system data for relabelled Cartan types¶
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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()
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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
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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)}
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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)}
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class
sage.combinat.root_system.type_relabel.
CartanType
(type, relabelling)¶ Bases:
sage.combinat.root_system.cartan_type.CartanType_decorator
A class for relabelled Cartan types.
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ascii_art
(label=<function <lambda>>, node=None)¶ Return 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
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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}
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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)]
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index_set
()¶ EXAMPLES:
sage: ct = CartanType(['G', 2]).relabel({1:2,2:1}) sage: ct.index_set() (1, 2)
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type
()¶ Return the type of
self
orNone
if unknown.EXAMPLES:
sage: ct = CartanType(['G', 2]).relabel({1:2,2:1}) sage: ct.type() 'G'
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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()
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basic_untwisted
()¶ Return the basic untwisted Cartan type associated with this affine Cartan type.
Given an affine type \(X_n^{(r)}\), the basic untwisted type is \(X_n\). In other words, it is the classical Cartan type that is twisted to obtain
self
.EXAMPLES:
sage: ct = CartanType(['A', 5, 2]).relabel({0:1, 1:0, 2:2, 3:3}) sage: ct.basic_untwisted() ['A', 5]
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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}
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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
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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
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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 typerelabelling
– 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 relabelledf(i)
(or, byf[i]
iff
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}
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AmbientSpace
¶ alias of
AmbientSpace
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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}
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}