# 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.
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')


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)]

class sage.combinat.root_system.dynkin_diagram.DynkinDiagram_class(t=None, index_set=None, **options)

A Dynkin diagram.

DynkinDiagram()

INPUT:

• t – a Cartan type, Cartan matrix, or None

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: 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.

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: list(sorted(d.edges()))
[(2, 3, 1), (3, 2, 1)]

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]

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]

cartan_type()

EXAMPLES:

sage: DynkinDiagram("A2","B2","F4").cartan_type()
A2xB2xF4

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)]

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()
[]

dynkin_diagram()

EXAMPLES:

sage: DynkinDiagram(['C',3]).dynkin_diagram()
O---O=<=O
1   2   3
C3

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)

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

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_crystalographic()
doctest:...: DeprecationWarning: is_crystalographic is deprecated. Please use is_crystallographic instead.
See http://trac.sagemath.org/14673 for details.
True

is_crystalographic(*args, **kwds)

Deprecated: Use is_crystallographic() instead. See trac ticket #14673 for details.

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

is_irreducible()

Check if self corresponds to an irreducible root system.

EXAMPLES:

sage: CartanType(['F',4]).dynkin_diagram().is_irreducible()
True

rank()

Returns the index set for this Dynkin diagram

EXAMPLES:

sage: DynkinDiagram(['C',3]).rank()
3
sage: DynkinDiagram("A2","B2","F4").rank()
8

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

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)]

symmetrizer()

Return the symmetrizer of the corresponding Cartan matrix.

EXAMPLES:

sage: d = DynkinDiagram()
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: 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]

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


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