Dynkin diagrams¶
AUTHORS:
- Travis Scrimshaw (2012-04-22): Nicolas M. Thiery moved Cartan matrix creation to here and I cached results for speed.
- Travis Scrimshaw (2013-06-11): Changed inputs of Dynkin diagrams to handle other Dynkin diagrams and graphs. Implemented remaining Cartan type methods.
- Christian Stump, Travis Scrimshaw (2013-04-11): Added Cartan matrix as possible input for Dynkin diagrams.
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sage.combinat.root_system.dynkin_diagram.
DynkinDiagram
(*args, **kwds)¶ Return the Dynkin diagram corresponding to the input.
INPUT:
The input can be one of the following:
- empty to obtain an empty Dynkin diagram
- a Cartan type
- a Cartan matrix
- a Cartan matrix and an indexing set
One can also input an indexing set by passing a tuple using the optional argument
index_set
.The edge multiplicities are encoded as edge labels. For the corresponding Cartan matrices, this uses the convention in Hong and Kang, Kac, Fulton and Harris, and crystals. This is the opposite convention in Bourbaki and Wikipedia’s Dynkin diagram (Wikipedia article Dynkin_diagram). That is for \(i \neq j\):
i <--k-- j <==> a_ij = -k <==> -scalar(coroot[i], root[j]) = k <==> multiple arrows point from the longer root to the shorter one
For example, in type \(C_2\), we have:
sage: C2 = DynkinDiagram(['C',2]); C2 O=<=O 1 2 C2 sage: C2.cartan_matrix() [ 2 -2] [-1 2]
However Bourbaki would have the Cartan matrix as:
\[\begin{split}\begin{bmatrix} 2 & -1 \\ -2 & 2 \end{bmatrix}.\end{split}\]EXAMPLES:
sage: DynkinDiagram(['A', 4]) O---O---O---O 1 2 3 4 A4 sage: DynkinDiagram(['A',1],['A',1]) O 1 O 2 A1xA1 sage: R = RootSystem("A2xB2xF4") sage: DynkinDiagram(R) O---O 1 2 O=>=O 3 4 O---O=>=O---O 5 6 7 8 A2xB2xF4 sage: R = RootSystem("A2xB2xF4") sage: CM = R.cartan_matrix(); CM [ 2 -1| 0 0| 0 0 0 0] [-1 2| 0 0| 0 0 0 0] [-----+-----+-----------] [ 0 0| 2 -1| 0 0 0 0] [ 0 0|-2 2| 0 0 0 0] [-----+-----+-----------] [ 0 0| 0 0| 2 -1 0 0] [ 0 0| 0 0|-1 2 -1 0] [ 0 0| 0 0| 0 -2 2 -1] [ 0 0| 0 0| 0 0 -1 2] sage: DD = DynkinDiagram(CM); DD O---O 1 2 O=>=O 3 4 O---O=>=O---O 5 6 7 8 A2xB2xF4 sage: DD.cartan_matrix() [ 2 -1 0 0 0 0 0 0] [-1 2 0 0 0 0 0 0] [ 0 0 2 -1 0 0 0 0] [ 0 0 -2 2 0 0 0 0] [ 0 0 0 0 2 -1 0 0] [ 0 0 0 0 -1 2 -1 0] [ 0 0 0 0 0 -2 2 -1] [ 0 0 0 0 0 0 -1 2]
We can also create Dynkin diagrams from arbitrary Cartan matrices:
sage: C = CartanMatrix([[2, -3], [-4, 2]]) sage: DynkinDiagram(C) Dynkin diagram of rank 2 sage: C.index_set() (0, 1) sage: CI = CartanMatrix([[2, -3], [-4, 2]], [3, 5]) sage: DI = DynkinDiagram(CI) sage: DI.index_set() (3, 5) sage: CII = CartanMatrix([[2, -3], [-4, 2]]) sage: DII = DynkinDiagram(CII, ('y', 'x')) sage: DII.index_set() ('x', 'y')
See also
CartanType()
for a general discussion on Cartan types and in particular node labeling conventions.TESTS:
Check that trac ticket #15277 is fixed by not having edges from 0’s:
sage: CM = CartanMatrix([[2,-1,0,0],[-3,2,-2,-2],[0,-1,2,-1],[0,-1,-1,2]]) sage: CM [ 2 -1 0 0] [-3 2 -2 -2] [ 0 -1 2 -1] [ 0 -1 -1 2] sage: CM.dynkin_diagram().edges() [(0, 1, 3), (1, 0, 1), (1, 2, 1), (1, 3, 1), (2, 1, 2), (2, 3, 1), (3, 1, 2), (3, 2, 1)]
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class
sage.combinat.root_system.dynkin_diagram.
DynkinDiagram_class
(t=None, index_set=None, **options)¶ Bases:
sage.graphs.digraph.DiGraph
,sage.combinat.root_system.cartan_type.CartanType_abstract
A Dynkin diagram.
See also
INPUT:
t
– a Cartan type, Cartan matrix, orNone
EXAMPLES:
sage: DynkinDiagram(['A', 3]) O---O---O 1 2 3 A3 sage: C = CartanMatrix([[2, -3], [-4, 2]]) sage: DynkinDiagram(C) Dynkin diagram of rank 2 sage: C.dynkin_diagram().cartan_matrix() == C True
TESTS:
Check that the correct type is returned when copied:
sage: d = DynkinDiagram(['A', 3]) sage: type(copy(d)) <class 'sage.combinat.root_system.dynkin_diagram.DynkinDiagram_class'>
We check that trac ticket #14655 is fixed:
sage: cd = copy(d) sage: cd.add_vertex(4) sage: d.vertices() != cd.vertices() True
Implementation note: if a Cartan type is given, then the nodes are initialized from the index set of this Cartan type.
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add_edge
(i, j, label=1)¶ EXAMPLES:
sage: from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class sage: d = DynkinDiagram_class(CartanType(['A',3])) sage: list(sorted(d.edges())) [] sage: d.add_edge(2, 3) sage: list(sorted(d.edges())) [(2, 3, 1), (3, 2, 1)]
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static
an_instance
()¶ Returns an example of Dynkin diagram
EXAMPLES:
sage: from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class sage: g = DynkinDiagram_class.an_instance() sage: g Dynkin diagram of rank 3 sage: g.cartan_matrix() [ 2 -1 -1] [-2 2 -1] [-1 -1 2]
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cartan_matrix
()¶ Returns the Cartan matrix for this Dynkin diagram
EXAMPLES:
sage: DynkinDiagram(['C',3]).cartan_matrix() [ 2 -1 0] [-1 2 -2] [ 0 -1 2]
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cartan_type
()¶ EXAMPLES:
sage: DynkinDiagram("A2","B2","F4").cartan_type() A2xB2xF4
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column
(j)¶ Returns the \(j^{th}\) column \((a_{i,j})_i\) of the Cartan matrix corresponding to this Dynkin diagram, as a container (or iterator) of tuples \((i, a_{i,j})\)
EXAMPLES:
sage: g = DynkinDiagram(["B",4]) sage: [ (i,a) for (i,a) in g.column(3) ] [(3, 2), (2, -1), (4, -2)]
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dual
()¶ Returns the dual Dynkin diagram, obtained by reversing all edges.
EXAMPLES:
sage: D = DynkinDiagram(['C',3]) sage: D.edges() [(1, 2, 1), (2, 1, 1), (2, 3, 1), (3, 2, 2)] sage: D.dual() O---O=>=O 1 2 3 B3 sage: D.dual().edges() [(1, 2, 1), (2, 1, 1), (2, 3, 2), (3, 2, 1)] sage: D.dual() == DynkinDiagram(['B',3]) True
TESTS:
sage: D = DynkinDiagram(['A',0]); D A0 sage: D.edges() [] sage: D.dual() A0 sage: D.dual().edges() [] sage: D = DynkinDiagram(['A',1]) sage: D.edges() [] sage: D.dual() O 1 A1 sage: D.dual().edges() []
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dynkin_diagram
()¶ EXAMPLES:
sage: DynkinDiagram(['C',3]).dynkin_diagram() O---O=<=O 1 2 3 C3
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index_set
()¶ EXAMPLES:
sage: DynkinDiagram(['C',3]).index_set() (1, 2, 3) sage: DynkinDiagram("A2","B2","F4").index_set() (1, 2, 3, 4, 5, 6, 7, 8)
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is_affine
()¶ Check if
self
corresponds to an affine root system.EXAMPLES:
sage: CartanType(['F',4]).dynkin_diagram().is_affine() False sage: D = DynkinDiagram(CartanMatrix([[2, -4], [-3, 2]])) sage: D.is_affine() False
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is_crystallographic
()¶ Implements
CartanType_abstract.is_crystallographic()
A Dynkin diagram always corresponds to a crystallographic root system.
EXAMPLES:
sage: CartanType(['F',4]).dynkin_diagram().is_crystallographic() True
TESTS:
sage: CartanType(['G',2]).dynkin_diagram().is_crystallographic() True
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is_finite
()¶ Check if
self
corresponds to a finite root system.EXAMPLES:
sage: CartanType(['F',4]).dynkin_diagram().is_finite() True sage: D = DynkinDiagram(CartanMatrix([[2, -4], [-3, 2]])) sage: D.is_finite() False
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is_irreducible
()¶ Check if
self
corresponds to an irreducible root system.EXAMPLES:
sage: CartanType(['F',4]).dynkin_diagram().is_irreducible() True
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rank
()¶ Returns the index set for this Dynkin diagram
EXAMPLES:
sage: DynkinDiagram(['C',3]).rank() 3 sage: DynkinDiagram("A2","B2","F4").rank() 8
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relabel
(relabelling, inplace=False, **kwds)¶ Return the relabelling Dynkin diagram of
self
.EXAMPLES:
sage: D = DynkinDiagram(['C',3]) sage: D.relabel({1:0, 2:4, 3:1}) O---O=<=O 0 4 1 C3 relabelled by {1: 0, 2: 4, 3: 1} sage: D O---O=<=O 1 2 3 C3
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row
(i)¶ Returns the \(i^{th}\) row \((a_{i,j})_j\) of the Cartan matrix corresponding to this Dynkin diagram, as a container (or iterator) of tuples \((j, a_{i,j})\)
EXAMPLES:
sage: g = DynkinDiagram(["C",4]) sage: [ (i,a) for (i,a) in g.row(3) ] [(3, 2), (2, -1), (4, -2)]
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subtype
(index_set)¶ Return a subtype of
self
given byindex_set
.A subtype can be considered the Dynkin diagram induced from the Dynkin diagram of
self
byindex_set
.EXAMPLES:
sage: D = DynkinDiagram(['A',6,2]); D O=<=O---O=<=O 0 1 2 3 BC3~ sage: D.subtype([1,2,3]) Dynkin diagram of rank 3
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symmetrizer
()¶ Return the symmetrizer of the corresponding Cartan matrix.
EXAMPLES:
sage: d = DynkinDiagram() sage: d.add_edge(1,2,3) sage: d.add_edge(2,3) sage: d.add_edge(3,4,3) sage: d.symmetrizer() Finite family {1: 9, 2: 3, 3: 3, 4: 1}
TESTS:
We check that trac ticket #15740 is fixed:
sage: d = DynkinDiagram() sage: d.add_edge(1,2,3) sage: d.add_edge(2,3) sage: d.add_edge(3,4,3) sage: L = d.root_system().root_lattice() sage: al = L.simple_roots() sage: al[1].associated_coroot() alphacheck[1] sage: al[1].reflection(al[2]) alpha[1] + 3*alpha[2]
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sage.combinat.root_system.dynkin_diagram.
precheck
(t, letter=None, length=None, affine=None, n_ge=None, n=None)¶ EXAMPLES:
sage: from sage.combinat.root_system.dynkin_diagram import precheck sage: ct = CartanType(['A',4]) sage: precheck(ct, letter='C') Traceback (most recent call last): ... ValueError: t[0] must be = 'C' sage: precheck(ct, affine=1) Traceback (most recent call last): ... ValueError: t[2] must be = 1 sage: precheck(ct, length=3) Traceback (most recent call last): ... ValueError: len(t) must be = 3 sage: precheck(ct, n=3) Traceback (most recent call last): ... ValueError: t[1] must be = 3 sage: precheck(ct, n_ge=5) Traceback (most recent call last): ... ValueError: t[1] must be >= 5