Cartan types#

Todo

Why does sphinx complain if I use sections here?

Introduction

Loosely speaking, Dynkin diagrams (or equivalently Cartan matrices) are graphs which are used to classify root systems, Coxeter and Weyl groups, Lie algebras, Lie groups, crystals, etc. up to an isomorphism. Cartan types are a standard set of names for those Dynkin diagrams (see Wikipedia article Dynkin_diagram).

Let us consider, for example, the Cartan type \(A_4\):

sage: T = CartanType(['A', 4]); T
['A', 4]

It is the name of the following Dynkin diagram:

sage: DynkinDiagram(T)                                                              # needs sage.graphs
O---O---O---O
1   2   3   4
A4

Note

For convenience, the following shortcuts are available:

sage: DynkinDiagram(['A',4])                                                    # needs sage.graphs
O---O---O---O
1   2   3   4
A4
sage: DynkinDiagram('A4')                                                       # needs sage.graphs
O---O---O---O
1   2   3   4
A4
sage: T.dynkin_diagram()                                                        # needs sage.graphs
O---O---O---O
1   2   3   4
A4

See DynkinDiagram for how to further manipulate Dynkin diagrams.

From this data (the Cartan datum), one can construct the associated root system:

sage: RootSystem(T)
Root system of type ['A', 4]

The associated Weyl group of \(A_n\) is the symmetric group \(S_{n+1}\):

sage: W = WeylGroup(T); W                                                           # needs sage.libs.gap
Weyl Group of type ['A', 4] (as a matrix group acting on the ambient space)
sage: W.cardinality()                                                               # needs sage.libs.gap
120

while the Lie algebra is \(sl_{n+1}\), and the Lie group \(SL_{n+1}\) (TODO: illustrate this once this is implemented).

One may also construct crystals associated to various Dynkin diagrams. For example:

sage: C = crystals.Letters(T); C                                                    # needs sage.combinat
The crystal of letters for type ['A', 4]
sage: C.list()                                                                      # needs sage.combinat
[1, 2, 3, 4, 5]

sage: C = crystals.Tableaux(T, shape=[2]); C                                        # needs sage.combinat
The crystal of tableaux of type ['A', 4] and shape(s) [[2]]
sage: C.cardinality()                                                               # needs sage.combinat
15

Here is a sample of all the finite irreducible crystallographic Cartan types:

sage: CartanType.samples(finite=True, crystallographic=True)
[['A', 1], ['A', 5], ['B', 1], ['B', 5], ['C', 1], ['C', 5], ['D', 2], ['D', 3], ['D', 5],
 ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2]]

One can also get latex representations of the crystallographic Cartan types and their corresponding Dynkin diagrams:

sage: [latex(ct) for ct in CartanType.samples(crystallographic=True)]
[A_{1}, A_{5}, B_{1}, B_{5}, C_{1}, C_{5}, D_{2}, D_{3}, D_{5},
 E_6, E_7, E_8, F_4, G_2,
 A_{1}^{(1)}, A_{5}^{(1)}, B_{1}^{(1)}, B_{5}^{(1)},
 C_{1}^{(1)}, C_{5}^{(1)}, D_{3}^{(1)}, D_{5}^{(1)},
 E_6^{(1)}, E_7^{(1)}, E_8^{(1)}, F_4^{(1)}, G_2^{(1)},
 BC_{1}^{(2)}, BC_{5}^{(2)},
 B_{5}^{(1)\vee}, C_{4}^{(1)\vee}, F_4^{(1)\vee},
 G_2^{(1)\vee}, BC_{1}^{(2)\vee}, BC_{5}^{(2)\vee}]
sage: view([DynkinDiagram(ct) for ct in CartanType.samples(crystallographic=True)]) # not tested

Non-crystallographic Cartan types are also partially supported:

sage: CartanType.samples(finite=True, crystallographic=False)
[['I', 5], ['H', 3], ['H', 4]]

In Sage, a Cartan type is used as a database of type-specific information and algorithms (see e.g. sage.combinat.root_system.type_A). This database includes how to construct the Dynkin diagram, the ambient space for the root system (see Wikipedia article Root_system), and further mathematical properties:

sage: T.is_finite(), T.is_simply_laced(), T.is_affine(), T.is_crystallographic()
(True, True, False, True)

In particular, a Sage Cartan type is endowed with a fixed choice of labels for the nodes of the Dynkin diagram. This choice follows the conventions of Nicolas Bourbaki, Lie Groups and Lie Algebras: Chapter 4-6, Elements of Mathematics, Springer (2002). ISBN 978-3540426509. For example:

sage: T = CartanType(['D', 4])
sage: DynkinDiagram(T)                                                              # needs sage.graphs
    O 4
    |
    |
O---O---O
1   2   3
D4

sage: E6 = CartanType(['E',6])
sage: DynkinDiagram(E6)                                                             # needs sage.graphs
        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6
E6

Note

The direction of the arrows is the opposite (i.e. the transpose) of Bourbaki’s convention, but agrees with Kac’s.

For example, in type \(C_2\), we have:

sage: C2 = DynkinDiagram(['C',2]); C2                                           # needs sage.graphs
O=<=O
1   2
C2
sage: C2.cartan_matrix()                                                        # needs sage.graphs
[ 2 -2]
[-1  2]

However Bourbaki would have the Cartan matrix as:

\[\begin{split}\begin{bmatrix} 2 & -1 \\ -2 & 2 \end{bmatrix}.\end{split}\]

If desired, other node labelling conventions can be achieved. For example the Kac labelling for type \(E_6\) can be obtained via:

sage: E6.relabel({1:1,2:6,3:2,4:3,5:4,6:5}).dynkin_diagram()                        # needs sage.graphs
        O 6
        |
        |
O---O---O---O---O
1   2   3   4   5
E6 relabelled by {1: 1, 2: 6, 3: 2, 4: 3, 5: 4, 6: 5}

Contributions implementing other conventions are very welcome.

Another option is to build from scratch a new Dynkin diagram. The architecture has been designed to make it fairly easy to add other labelling conventions. In particular, we strived at choosing type free algorithms whenever possible, so in principle most features should remain available even with custom Cartan types. This has not been used much yet, so some rough corners certainly remain.

Here, we construct the hyperbolic example of Exercise 4.9 p. 57 of Kac, Infinite Dimensional Lie Algebras. We start with an empty Dynkin diagram, and add a couple nodes:

sage: g = DynkinDiagram()                                                           # needs sage.graphs
sage: g.add_vertices([1,2,3])                                                       # needs sage.graphs

Note that the diagonal of the Cartan matrix is already initialized:

sage: g.cartan_matrix()                                                             # needs sage.graphs
[2 0 0]
[0 2 0]
[0 0 2]

Then we add a couple edges:

sage: g.add_edge(1,2,2)                                                             # needs sage.graphs
sage: g.add_edge(1,3)                                                               # needs sage.graphs
sage: g.add_edge(2,3)                                                               # needs sage.graphs

and we get the desired Cartan matrix:

sage: g.cartan_matrix()                                                             # needs sage.graphs
[2 0 0]
[0 2 0]
[0 0 2]

Oops, the Cartan matrix did not change! This is because it is cached for efficiency (see cached_method). In general, a Dynkin diagram should not be modified after having been used.

Warning

this is not checked currently

Todo

add a method set_mutable() as, say, for matrices

Here, we can work around this by clearing the cache:

sage: delattr(g, 'cartan_matrix')                                                   # needs sage.graphs

Now we get the desired Cartan matrix:

sage: g.cartan_matrix()                                                             # needs sage.graphs
[ 2 -1 -1]
[-2  2 -1]
[-1 -1  2]

Note that backward edges have been automatically added:

sage: g.edges(sort=True)                                                            # needs sage.graphs
[(1, 2, 2), (1, 3, 1), (2, 1, 1), (2, 3, 1), (3, 1, 1), (3, 2, 1)]

Reducible Cartan types

Reducible Cartan types can be specified by passing a sequence or list of irreducible Cartan types:

sage: CartanType(['A',2],['B',2])
A2xB2
sage: CartanType([['A',2],['B',2]])
A2xB2
sage: CartanType(['A',2],['B',2]).is_reducible()
True

or using the following short hand notation:

sage: CartanType("A2xB2")
A2xB2
sage: CartanType("A2","B2") == CartanType("A2xB2")
True

Degenerate cases

When possible, type \(I_n\) is automatically converted to the isomorphic crystallographic Cartan types (any reason not to do so?):

sage: CartanType(["I",1])
A1xA1
sage: CartanType(["I",3])
['A', 2]
sage: CartanType(["I",4])
['C', 2]
sage: CartanType(["I",6])
['G', 2]

The Dynkin diagrams for types \(B_1\), \(C_1\), \(D_2\), and \(D_3\) are isomorphic to that for \(A_1\), \(A_1\), \(A_1 \times A_1\), and \(A_3\), respectively. However their natural ambient space realizations (stemming from the corresponding infinite families of Lie groups) are different. Therefore, the Cartan types are considered as distinct:

sage: CartanType(['B',1])
['B', 1]
sage: CartanType(['C',1])
['C', 1]
sage: CartanType(['D',2])
['D', 2]
sage: CartanType(['D',3])
['D', 3]

Affine Cartan types

For affine types, we use the usual conventions for affine Coxeter groups: each affine type is either untwisted (that is arise from the natural affinisation of a finite Cartan type):

sage: CartanType(["A", 4, 1]).dynkin_diagram()                                      # needs sage.graphs
0
O-----------+
|           |
|           |
O---O---O---O
1   2   3   4
A4~
sage: CartanType(["B", 4, 1]).dynkin_diagram()                                      # needs sage.graphs
    O 0
    |
    |
O---O---O=>=O
1   2   3   4
B4~

or dual thereof:

sage: CartanType(["B", 4, 1]).dual().dynkin_diagram()                               # needs sage.graphs
    O 0
    |
    |
O---O---O=<=O
1   2   3   4
B4~*

or is of type \(\widetilde{BC}_n\) (which yields an irreducible, but nonreduced root system):

sage: CartanType(["BC", 4, 2]).dynkin_diagram()                                     # needs sage.graphs
O=<=O---O---O=<=O
0   1   2   3   4
BC4~

This includes the two degenerate cases:

sage: CartanType(["A", 1, 1]).dynkin_diagram()                                      # needs sage.graphs
O<=>O
0   1
A1~
sage: CartanType(["BC", 1, 2]).dynkin_diagram()                                     # needs sage.graphs
  4
O=<=O
0   1
BC1~

For the user convenience, Kac’s notations for twisted affine types are automatically translated into the previous ones:

sage: # needs sage.graphs
sage: CartanType(["A", 9, 2])
['B', 5, 1]^*
sage: CartanType(["A", 9, 2]).dynkin_diagram()
    O 0
    |
    |
O---O---O---O=<=O
1   2   3   4   5
B5~*
sage: CartanType(["A", 10, 2]).dynkin_diagram()
O=<=O---O---O---O=<=O
0   1   2   3   4   5
BC5~
sage: CartanType(["D", 5, 2]).dynkin_diagram()
O=<=O---O---O=>=O
0   1   2   3   4
C4~*
sage: CartanType(["D", 4, 3]).dynkin_diagram()
  3
O=>=O---O
2   1   0
G2~* relabelled by {0: 0, 1: 2, 2: 1}
sage: CartanType(["E", 6, 2]).dynkin_diagram()
O---O---O=<=O---O
0   1   2   3   4
F4~*

Additionally one can set the notation option to use Kac’s notation:

sage: # needs sage.graphs
sage: CartanType.options['notation'] = 'Kac'
sage: CartanType(["A", 9, 2])
['A', 9, 2]
sage: CartanType(["A", 9, 2]).dynkin_diagram()
    O 0
    |
    |
O---O---O---O=<=O
1   2   3   4   5
A9^2
sage: CartanType(["A", 10, 2]).dynkin_diagram()
O=<=O---O---O---O=<=O
0   1   2   3   4   5
A10^2
sage: CartanType(["D", 5, 2]).dynkin_diagram()
O=<=O---O---O=>=O
0   1   2   3   4
D5^2
sage: CartanType(["D", 4, 3]).dynkin_diagram()
  3
O=>=O---O
2   1   0
D4^3
sage: CartanType(["E", 6, 2]).dynkin_diagram()
O---O---O=<=O---O
0   1   2   3   4
E6^2
sage: CartanType.options['notation'] = 'BC'

Infinite Cartan types

There are minimal implementations of the Cartan types \(A_{\infty}\) and \(A_{+\infty}\). In sage \(oo\) is the same as \(+Infinity\), so \(NN\) and \(ZZ\) are used to differentiate between the \(A_{+\infty}\) and \(A_{\infty}\) root systems:

sage: CartanType(['A', NN])
['A', NN]
sage: print(CartanType(['A', NN]).ascii_art())
O---O---O---O---O---O---O---..
0   1   2   3   4   5   6
sage: CartanType(['A', ZZ])
['A', ZZ]
sage: print(CartanType(['A', ZZ]).ascii_art())
..---O---O---O---O---O---O---O---..
    -3  -2  -1   0   1   2   3

There are also the following shorthands:

sage: CartanType("Aoo")
['A', ZZ]
sage: CartanType("A+oo")
['A', NN]

Abstract classes for Cartan types

Concrete classes for Cartan types

CartanType_standard
CartanType_standard_finite
CartanType_standard_affine
CartanType_standard_untwisted_affine

Type specific data

The data essentially consists of a description of the Dynkin/Coxeter diagram and, when relevant, of the natural embedding of the root system in an Euclidean space. Everything else is reconstructed from this data.

Todo

Should those indexes come before the introduction?

sage.combinat.root_system.cartan_type.CartanType(*args)#

Cartan types

Todo

Why does sphinx complain if I use sections here?

Introduction

Let us consider, for example, the Cartan type \(A_4\):

sage: T = CartanType(['A', 4]); T
['A', 4]

It is the name of the following Dynkin diagram:

sage: DynkinDiagram(T)                                                              # needs sage.graphs
O---O---O---O
1   2   3   4
A4

Note

For convenience, the following shortcuts are available:

sage: DynkinDiagram(['A',4])                                                    # needs sage.graphs
O---O---O---O
1   2   3   4
A4
sage: DynkinDiagram('A4')                                                       # needs sage.graphs
O---O---O---O
1   2   3   4
A4
sage: T.dynkin_diagram()                                                        # needs sage.graphs
O---O---O---O
1   2   3   4
A4

See DynkinDiagram for how to further manipulate Dynkin diagrams.

From this data (the Cartan datum), one can construct the associated root system:

sage: RootSystem(T)
Root system of type ['A', 4]

The associated Weyl group of \(A_n\) is the symmetric group \(S_{n+1}\):

sage: W = WeylGroup(T); W                                                           # needs sage.libs.gap
Weyl Group of type ['A', 4] (as a matrix group acting on the ambient space)
sage: W.cardinality()                                                               # needs sage.libs.gap
120

while the Lie algebra is \(sl_{n+1}\), and the Lie group \(SL_{n+1}\) (TODO: illustrate this once this is implemented).

One may also construct crystals associated to various Dynkin diagrams. For example:

sage: C = crystals.Letters(T); C                                                    # needs sage.combinat
The crystal of letters for type ['A', 4]
sage: C.list()                                                                      # needs sage.combinat
[1, 2, 3, 4, 5]

sage: C = crystals.Tableaux(T, shape=[2]); C                                        # needs sage.combinat
The crystal of tableaux of type ['A', 4] and shape(s) [[2]]
sage: C.cardinality()                                                               # needs sage.combinat
15

Here is a sample of all the finite irreducible crystallographic Cartan types:

sage: CartanType.samples(finite=True, crystallographic=True)
[['A', 1], ['A', 5], ['B', 1], ['B', 5], ['C', 1], ['C', 5], ['D', 2], ['D', 3], ['D', 5],
 ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2]]

One can also get latex representations of the crystallographic Cartan types and their corresponding Dynkin diagrams:

sage: [latex(ct) for ct in CartanType.samples(crystallographic=True)]
[A_{1}, A_{5}, B_{1}, B_{5}, C_{1}, C_{5}, D_{2}, D_{3}, D_{5},
 E_6, E_7, E_8, F_4, G_2,
 A_{1}^{(1)}, A_{5}^{(1)}, B_{1}^{(1)}, B_{5}^{(1)},
 C_{1}^{(1)}, C_{5}^{(1)}, D_{3}^{(1)}, D_{5}^{(1)},
 E_6^{(1)}, E_7^{(1)}, E_8^{(1)}, F_4^{(1)}, G_2^{(1)},
 BC_{1}^{(2)}, BC_{5}^{(2)},
 B_{5}^{(1)\vee}, C_{4}^{(1)\vee}, F_4^{(1)\vee},
 G_2^{(1)\vee}, BC_{1}^{(2)\vee}, BC_{5}^{(2)\vee}]
sage: view([DynkinDiagram(ct) for ct in CartanType.samples(crystallographic=True)]) # not tested

Non-crystallographic Cartan types are also partially supported:

sage: CartanType.samples(finite=True, crystallographic=False)
[['I', 5], ['H', 3], ['H', 4]]

sage: T.is_finite(), T.is_simply_laced(), T.is_affine(), T.is_crystallographic()
(True, True, False, True)

sage: T = CartanType(['D', 4])
sage: DynkinDiagram(T)                                                              # needs sage.graphs
    O 4
    |
    |
O---O---O
1   2   3
D4

sage: E6 = CartanType(['E',6])
sage: DynkinDiagram(E6)                                                             # needs sage.graphs
        O 2
        |
        |
O---O---O---O---O
1   3   4   5   6
E6

Note

The direction of the arrows is the opposite (i.e. the transpose) of Bourbaki’s convention, but agrees with Kac’s.

For example, in type \(C_2\), we have:

sage: C2 = DynkinDiagram(['C',2]); C2                                           # needs sage.graphs
O=<=O
1   2
C2
sage: C2.cartan_matrix()                                                        # needs sage.graphs
[ 2 -2]
[-1  2]

However Bourbaki would have the Cartan matrix as:

\[\begin{split}\begin{bmatrix} 2 & -1 \\ -2 & 2 \end{bmatrix}.\end{split}\]

If desired, other node labelling conventions can be achieved. For example the Kac labelling for type \(E_6\) can be obtained via:

sage: E6.relabel({1:1,2:6,3:2,4:3,5:4,6:5}).dynkin_diagram()                        # needs sage.graphs
        O 6
        |
        |
O---O---O---O---O
1   2   3   4   5
E6 relabelled by {1: 1, 2: 6, 3: 2, 4: 3, 5: 4, 6: 5}

Contributions implementing other conventions are very welcome.

Here, we construct the hyperbolic example of Exercise 4.9 p. 57 of Kac, Infinite Dimensional Lie Algebras. We start with an empty Dynkin diagram, and add a couple nodes:

sage: g = DynkinDiagram()                                                           # needs sage.graphs
sage: g.add_vertices([1,2,3])                                                       # needs sage.graphs

Note that the diagonal of the Cartan matrix is already initialized:

sage: g.cartan_matrix()                                                             # needs sage.graphs
[2 0 0]
[0 2 0]
[0 0 2]

Then we add a couple edges:

sage: g.add_edge(1,2,2)                                                             # needs sage.graphs
sage: g.add_edge(1,3)                                                               # needs sage.graphs
sage: g.add_edge(2,3)                                                               # needs sage.graphs

and we get the desired Cartan matrix:

sage: g.cartan_matrix()                                                             # needs sage.graphs
[2 0 0]
[0 2 0]
[0 0 2]

Oops, the Cartan matrix did not change! This is because it is cached for efficiency (see cached_method). In general, a Dynkin diagram should not be modified after having been used.

Warning

this is not checked currently

Todo

add a method set_mutable() as, say, for matrices

Here, we can work around this by clearing the cache:

sage: delattr(g, 'cartan_matrix')                                                   # needs sage.graphs

Now we get the desired Cartan matrix:

sage: g.cartan_matrix()                                                             # needs sage.graphs
[ 2 -1 -1]
[-2  2 -1]
[-1 -1  2]

Note that backward edges have been automatically added:

sage: g.edges(sort=True)                                                            # needs sage.graphs
[(1, 2, 2), (1, 3, 1), (2, 1, 1), (2, 3, 1), (3, 1, 1), (3, 2, 1)]

Reducible Cartan types

Reducible Cartan types can be specified by passing a sequence or list of irreducible Cartan types:

sage: CartanType(['A',2],['B',2])
A2xB2
sage: CartanType([['A',2],['B',2]])
A2xB2
sage: CartanType(['A',2],['B',2]).is_reducible()
True

or using the following short hand notation:

sage: CartanType("A2xB2")
A2xB2
sage: CartanType("A2","B2") == CartanType("A2xB2")
True

Degenerate cases

When possible, type \(I_n\) is automatically converted to the isomorphic crystallographic Cartan types (any reason not to do so?):

sage: CartanType(["I",1])
A1xA1
sage: CartanType(["I",3])
['A', 2]
sage: CartanType(["I",4])
['C', 2]
sage: CartanType(["I",6])
['G', 2]

sage: CartanType(['B',1])
['B', 1]
sage: CartanType(['C',1])
['C', 1]
sage: CartanType(['D',2])
['D', 2]
sage: CartanType(['D',3])
['D', 3]

Affine Cartan types

For affine types, we use the usual conventions for affine Coxeter groups: each affine type is either untwisted (that is arise from the natural affinisation of a finite Cartan type):

sage: CartanType(["A", 4, 1]).dynkin_diagram()                                      # needs sage.graphs
0
O-----------+
|           |
|           |
O---O---O---O
1   2   3   4
A4~
sage: CartanType(["B", 4, 1]).dynkin_diagram()                                      # needs sage.graphs
    O 0
    |
    |
O---O---O=>=O
1   2   3   4
B4~

or dual thereof:

sage: CartanType(["B", 4, 1]).dual().dynkin_diagram()                               # needs sage.graphs
    O 0
    |
    |
O---O---O=<=O
1   2   3   4
B4~*

or is of type \(\widetilde{BC}_n\) (which yields an irreducible, but nonreduced root system):

sage: CartanType(["BC", 4, 2]).dynkin_diagram()                                     # needs sage.graphs
O=<=O---O---O=<=O
0   1   2   3   4
BC4~

This includes the two degenerate cases:

sage: CartanType(["A", 1, 1]).dynkin_diagram()                                      # needs sage.graphs
O<=>O
0   1
A1~
sage: CartanType(["BC", 1, 2]).dynkin_diagram()                                     # needs sage.graphs
  4
O=<=O
0   1
BC1~

For the user convenience, Kac’s notations for twisted affine types are automatically translated into the previous ones:

sage: # needs sage.graphs
sage: CartanType(["A", 9, 2])
['B', 5, 1]^*
sage: CartanType(["A", 9, 2]).dynkin_diagram()
    O 0
    |
    |
O---O---O---O=<=O
1   2   3   4   5
B5~*
sage: CartanType(["A", 10, 2]).dynkin_diagram()
O=<=O---O---O---O=<=O
0   1   2   3   4   5
BC5~
sage: CartanType(["D", 5, 2]).dynkin_diagram()
O=<=O---O---O=>=O
0   1   2   3   4
C4~*
sage: CartanType(["D", 4, 3]).dynkin_diagram()
  3
O=>=O---O
2   1   0
G2~* relabelled by {0: 0, 1: 2, 2: 1}
sage: CartanType(["E", 6, 2]).dynkin_diagram()
O---O---O=<=O---O
0   1   2   3   4
F4~*

Additionally one can set the notation option to use Kac’s notation:

sage: # needs sage.graphs
sage: CartanType.options['notation'] = 'Kac'
sage: CartanType(["A", 9, 2])
['A', 9, 2]
sage: CartanType(["A", 9, 2]).dynkin_diagram()
    O 0
    |
    |
O---O---O---O=<=O
1   2   3   4   5
A9^2
sage: CartanType(["A", 10, 2]).dynkin_diagram()
O=<=O---O---O---O=<=O
0   1   2   3   4   5
A10^2
sage: CartanType(["D", 5, 2]).dynkin_diagram()
O=<=O---O---O=>=O
0   1   2   3   4
D5^2
sage: CartanType(["D", 4, 3]).dynkin_diagram()
  3
O=>=O---O
2   1   0
D4^3
sage: CartanType(["E", 6, 2]).dynkin_diagram()
O---O---O=<=O---O
0   1   2   3   4
E6^2
sage: CartanType.options['notation'] = 'BC'

Infinite Cartan types

sage: CartanType(['A', NN])
['A', NN]
sage: print(CartanType(['A', NN]).ascii_art())
O---O---O---O---O---O---O---..
0   1   2   3   4   5   6
sage: CartanType(['A', ZZ])
['A', ZZ]
sage: print(CartanType(['A', ZZ]).ascii_art())
..---O---O---O---O---O---O---O---..
    -3  -2  -1   0   1   2   3

There are also the following shorthands:

sage: CartanType("Aoo")
['A', ZZ]
sage: CartanType("A+oo")
['A', NN]

Abstract classes for Cartan types

Concrete classes for Cartan types

CartanType_standard
CartanType_standard_finite
CartanType_standard_affine
CartanType_standard_untwisted_affine

Type specific data

Todo

Should those indexes come before the introduction?

class sage.combinat.root_system.cartan_type.CartanTypeFactory#

Bases: SageObject

classmethod color(i)#

Default color scheme for the vertices of a Dynkin diagram (and associated objects)

EXAMPLES:

sage: CartanType.color(1)
'blue'
sage: CartanType.color(2)
'red'
sage: CartanType.color(3)
'green'

The default color is black:

sage: CartanType.color(0)
'black'

Negative indices get the same color as their positive counterparts:

sage: CartanType.color(-1)
'blue'
sage: CartanType.color(-2)
'red'
sage: CartanType.color(-3)
'green'

options = Current options for CartanType - dual_latex: \vee - dual_str: * - latex_marked: True - latex_relabel: True - mark_special_node: none - marked_node_str: X - notation: Stembridge - special_node_str: @#

samples(finite=None, affine=None, crystallographic=None)#

Return a sample of the available Cartan types.

INPUT:

finite – a boolean or None (default: None)
affine – a boolean or None (default: None)
crystallographic – a boolean or None (default: None)

The sample contains all the exceptional finite and affine Cartan types, as well as typical representatives of the infinite families.

EXAMPLES:

sage: CartanType.samples()
[['A', 1], ['A', 5], ['B', 1], ['B', 5], ['C', 1], ['C', 5], ['D', 2], ['D', 3], ['D', 5],
 ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2], ['I', 5], ['H', 3], ['H', 4],
 ['A', 1, 1], ['A', 5, 1], ['B', 1, 1], ['B', 5, 1],
 ['C', 1, 1], ['C', 5, 1], ['D', 3, 1], ['D', 5, 1],
 ['E', 6, 1], ['E', 7, 1], ['E', 8, 1], ['F', 4, 1], ['G', 2, 1], ['BC', 1, 2], ['BC', 5, 2],
 ['B', 5, 1]^*, ['C', 4, 1]^*, ['F', 4, 1]^*, ['G', 2, 1]^*, ['BC', 1, 2]^*, ['BC', 5, 2]^*]

The finite, affine and crystallographic options allow respectively for restricting to (non) finite, (non) affine, and (non) crystallographic Cartan types:

sage: CartanType.samples(finite=True)
[['A', 1], ['A', 5], ['B', 1], ['B', 5], ['C', 1], ['C', 5], ['D', 2], ['D', 3], ['D', 5],
 ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2], ['I', 5], ['H', 3], ['H', 4]]

sage: CartanType.samples(affine=True)
[['A', 1, 1], ['A', 5, 1], ['B', 1, 1], ['B', 5, 1],
 ['C', 1, 1], ['C', 5, 1], ['D', 3, 1], ['D', 5, 1],
 ['E', 6, 1], ['E', 7, 1], ['E', 8, 1], ['F', 4, 1], ['G', 2, 1], ['BC', 1, 2], ['BC', 5, 2],
 ['B', 5, 1]^*, ['C', 4, 1]^*, ['F', 4, 1]^*, ['G', 2, 1]^*, ['BC', 1, 2]^*, ['BC', 5, 2]^*]

sage: CartanType.samples(crystallographic=True)
[['A', 1], ['A', 5], ['B', 1], ['B', 5], ['C', 1], ['C', 5], ['D', 2], ['D', 3], ['D', 5],
 ['E', 6], ['E', 7], ['E', 8], ['F', 4], ['G', 2],
 ['A', 1, 1], ['A', 5, 1], ['B', 1, 1], ['B', 5, 1],
 ['C', 1, 1], ['C', 5, 1], ['D', 3, 1], ['D', 5, 1],
 ['E', 6, 1], ['E', 7, 1], ['E', 8, 1], ['F', 4, 1], ['G', 2, 1], ['BC', 1, 2], ['BC', 5, 2],
 ['B', 5, 1]^*, ['C', 4, 1]^*, ['F', 4, 1]^*, ['G', 2, 1]^*, ['BC', 1, 2]^*, ['BC', 5, 2]^*]

sage: CartanType.samples(crystallographic=False)
[['I', 5], ['H', 3], ['H', 4]]

Todo

add some reducible Cartan types (suggestions?)

class sage.combinat.root_system.cartan_type.CartanType_abstract#

Bases: object

Abstract class for Cartan types

Subclasses should implement:

dynkin_diagram()
cartan_matrix()
is_finite()
is_affine()
is_irreducible()

as_folding(folding_of=None, sigma=None)#

Return self realized as a folded Cartan type.

For finite and affine types, this is realized by the Dynkin diagram foldings:

\[\begin{split}\begin{array}{ccl} C_n^{(1)}, A_{2n}^{(2)}, A_{2n}^{(2)\dagger}, D_{n+1}^{(2)} & \hookrightarrow & A_{2n-1}^{(1)}, \\ A_{2n-1}^{(2)}, B_n^{(1)} & \hookrightarrow & D_{n+1}^{(1)}, \\ E_6^{(2)}, F_4^{(1)} & \hookrightarrow & E_6^{(1)}, \\ D_4^{(3)}, G_2^{(1)} & \hookrightarrow & D_4^{(1)}, \\ C_n & \hookrightarrow & A_{2n-1}, \\ B_n & \hookrightarrow & D_{n+1}, \\ F_4 & \hookrightarrow & E_6, \\ G_2 & \hookrightarrow & D_4. \end{array}\end{split}\]

For general types, this returns self as a folded type of self with \(\sigma\) as the identity map.

For more information on these foldings and folded Cartan types, see sage.combinat.root_system.type_folded.CartanTypeFolded.

If the optional inputs folding_of and sigma are specified, then this returns the folded Cartan type of self in folding_of given by the automorphism sigma.

EXAMPLES:

sage: CartanType(['B', 3, 1]).as_folding()
['B', 3, 1] as a folding of  ['D', 4, 1]
sage: CartanType(['F', 4]).as_folding()
['F', 4] as a folding of  ['E', 6]
sage: CartanType(['BC', 3, 2]).as_folding()
['BC', 3, 2] as a folding of  ['A', 5, 1]
sage: CartanType(['D', 4, 3]).as_folding()
['G', 2, 1]^* relabelled by {0: 0, 1: 2, 2: 1} as a folding of ['D', 4, 1]

coxeter_diagram()#

Return the Coxeter diagram for self.

EXAMPLES:

sage: # needs sage.graphs
sage: CartanType(['B',3]).coxeter_diagram()
Graph on 3 vertices
sage: CartanType(['A',3]).coxeter_diagram().edges(sort=True)
[(1, 2, 3), (2, 3, 3)]
sage: CartanType(['B',3]).coxeter_diagram().edges(sort=True)
[(1, 2, 3), (2, 3, 4)]
sage: CartanType(['G',2]).coxeter_diagram().edges(sort=True)
[(1, 2, 6)]
sage: CartanType(['F',4]).coxeter_diagram().edges(sort=True)
[(1, 2, 3), (2, 3, 4), (3, 4, 3)]

coxeter_matrix()#

Return the Coxeter matrix for self.

EXAMPLES:

sage: CartanType(['A', 4]).coxeter_matrix()                                 # needs sage.graphs
[1 3 2 2]
[3 1 3 2]
[2 3 1 3]
[2 2 3 1]

coxeter_type()#

Return the Coxeter type for self.

EXAMPLES:

sage: CartanType(['A', 4]).coxeter_type()
Coxeter type of ['A', 4]

dual()#

Return the dual Cartan type, possibly just as a formal dual.

EXAMPLES:

sage: CartanType(['A',3]).dual()
['A', 3]
sage: CartanType(["B", 3]).dual()
['C', 3]
sage: CartanType(['C',2]).dual()
['B', 2]
sage: CartanType(['D',4]).dual()
['D', 4]
sage: CartanType(['E',8]).dual()
['E', 8]
sage: CartanType(['F',4]).dual()
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}

index_set()#

Return the index set for self.

This is the list of the nodes of the associated Coxeter or Dynkin diagram.

EXAMPLES:

sage: CartanType(['A', 3, 1]).index_set()
(0, 1, 2, 3)
sage: CartanType(['D', 4]).index_set()
(1, 2, 3, 4)
sage: CartanType(['A', 7, 2]).index_set()
(0, 1, 2, 3, 4)
sage: CartanType(['A', 7, 2]).index_set()
(0, 1, 2, 3, 4)
sage: CartanType(['A', 6, 2]).index_set()
(0, 1, 2, 3)
sage: CartanType(['D', 6, 2]).index_set()
(0, 1, 2, 3, 4, 5)
sage: CartanType(['E', 6, 1]).index_set()
(0, 1, 2, 3, 4, 5, 6)
sage: CartanType(['E', 6, 2]).index_set()
(0, 1, 2, 3, 4)
sage: CartanType(['A', 2, 2]).index_set()
(0, 1)
sage: CartanType(['G', 2, 1]).index_set()
(0, 1, 2)
sage: CartanType(['F', 4, 1]).index_set()
(0, 1, 2, 3, 4)

is_affine()#

Return whether self is affine.

EXAMPLES:

sage: CartanType(['A', 3]).is_affine()
False
sage: CartanType(['A', 3, 1]).is_affine()
True

is_atomic()#

This method is usually equivalent to is_reducible(), except for the Cartan type \(D_2\).

\(D_2\) is not a standard Cartan type. It is equivalent to type \(A_1 \times A_1\) which is reducible; however the isomorphism from its ambient space (for the orthogonal group of degree 4) to that of \(A_1 \times A_1\) is non trivial, and it is useful to have it.

From a programming point of view its implementation is more similar to the irreducible types, and so the method is_atomic() is supplied.

EXAMPLES:

sage: CartanType("D2").is_atomic()
True
sage: CartanType("D2").is_irreducible()
False

is_compound()#: A short hand for not is_atomic().

is_crystallographic()#

Return whether this Cartan type is crystallographic.

This returns False by default. Derived class should override this appropriately.

EXAMPLES:

sage: [ [t, t.is_crystallographic() ] for t in CartanType.samples(finite=True) ]
[[['A', 1], True], [['A', 5], True],
 [['B', 1], True], [['B', 5], True],
 [['C', 1], True], [['C', 5], True],
 [['D', 2], True], [['D', 3], True], [['D', 5], True],
 [['E', 6], True], [['E', 7], True], [['E', 8], True],
 [['F', 4], True], [['G', 2], True],
 [['I', 5], False], [['H', 3], False], [['H', 4], False]]

is_finite()#

Return whether this Cartan type is finite.

EXAMPLES:

sage: from sage.combinat.root_system.cartan_type import CartanType_abstract
sage: C = CartanType_abstract()
sage: C.is_finite()
Traceback (most recent call last):
...
NotImplementedError: <abstract method is_finite at ...>

sage: CartanType(['A',4]).is_finite()
True
sage: CartanType(['A',4, 1]).is_finite()
False

is_implemented()#

Check whether the Cartan datum for self is actually implemented.

EXAMPLES:

sage: CartanType(["A",4,1]).is_implemented()                                # needs sage.graphs
True
sage: CartanType(['H',3]).is_implemented()
True

is_irreducible()#

Report whether this Cartan type is irreducible (i.e. simple). This should be overridden in any subclass.

This returns False by default. Derived class should override this appropriately.

EXAMPLES:

sage: from sage.combinat.root_system.cartan_type import CartanType_abstract
sage: C = CartanType_abstract()
sage: C.is_irreducible()
False

is_reducible()#

Report whether the root system is reducible (i.e. not simple), that is whether it can be factored as a product of root systems.

EXAMPLES:

sage: CartanType("A2xB3").is_reducible()
True
sage: CartanType(['A',2]).is_reducible()
False

is_simply_laced()#

Return whether this Cartan type is simply laced.

This returns False by default. Derived class should override this appropriately.

EXAMPLES:

sage: [ [t, t.is_simply_laced() ] for t in CartanType.samples() ]
[[['A', 1], True], [['A', 5], True],
 [['B', 1], True], [['B', 5], False],
 [['C', 1], True], [['C', 5], False],
 [['D', 2], True], [['D', 3], True], [['D', 5], True],
 [['E', 6], True], [['E', 7], True], [['E', 8], True],
 [['F', 4], False], [['G', 2], False], [['I', 5], False],
 [['H', 3], False], [['H', 4], False],
 [['A', 1, 1], False], [['A', 5, 1], True],
 [['B', 1, 1], False], [['B', 5, 1], False],
 [['C', 1, 1], False], [['C', 5, 1], False],
 [['D', 3, 1], True], [['D', 5, 1], True],
 [['E', 6, 1], True], [['E', 7, 1], True], [['E', 8, 1], True],
 [['F', 4, 1], False], [['G', 2, 1], False],
 [['BC', 1, 2], False], [['BC', 5, 2], False],
 [['B', 5, 1]^*, False], [['C', 4, 1]^*, False],
 [['F', 4, 1]^*, False], [['G', 2, 1]^*, False],
 [['BC', 1, 2]^*, False], [['BC', 5, 2]^*, False]]

marked_nodes(marked_nodes)#

Return a Cartan type with the nodes marked_nodes marked.

INPUT:

marked_nodes – a list of nodes to mark

EXAMPLES:

sage: CartanType(['F',4]).marked_nodes([1, 3]).dynkin_diagram()             # needs sage.graphs
X---O=>=X---O
1   2   3   4
F4 with nodes (1, 3) marked

options = Current options for CartanType - dual_latex: \vee - dual_str: * - latex_marked: True - latex_relabel: True - mark_special_node: none - marked_node_str: X - notation: Stembridge - special_node_str: @#

rank()#

Return the rank of self.

This is the number of nodes of the associated Coxeter or Dynkin diagram.

EXAMPLES:

sage: CartanType(['A', 4]).rank()
4
sage: CartanType(['A', 7, 2]).rank()
5
sage: CartanType(['I', 8]).rank()
2

relabel(relabelling)#

Return a relabelled copy of this Cartan type.

INPUT:

relabelling – a function (or a list or dictionary)

OUTPUT:

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:

sage: CartanType(['F',4]).relabel({ 1:4, 2:3, 3:2, 4:1 }).dynkin_diagram()   # needs sage.graphs
O---O=>=O---O
4   3   2   1
F4 relabelled by {1: 4, 2: 3, 3: 2, 4: 1}

root_system()#

Return the root system associated to self.

EXAMPLES:

sage: CartanType(['A',4]).root_system()
Root system of type ['A', 4]

subtype(index_set)#

Return a subtype of self given by index_set.

A subtype can be considered the Dynkin diagram induced from the Dynkin diagram of self by index_set.

EXAMPLES:

sage: ct = CartanType(['A',6,2])
sage: ct.dynkin_diagram()                                                   # needs sage.graphs
O=<=O---O=<=O
0   1   2   3
BC3~
sage: ct.subtype([1,2,3])                                                   # needs sage.graphs
['C', 3]

type()#

Return the type of self, or None if unknown.

This method should be overridden in any subclass.

EXAMPLES:

sage: from sage.combinat.root_system.cartan_type import CartanType_abstract
sage: C = CartanType_abstract()
sage: C.type() is None
True

class sage.combinat.root_system.cartan_type.CartanType_affine#

Bases: CartanType_simple, CartanType_crystallographic

An abstract class for simple affine Cartan types

AmbientSpace#: alias of AmbientSpace

a()#

Return the unique minimal non trivial annihilating linear combination of \(\alpha^\vee_0, \alpha^\vee, \ldots, \alpha^\vee\) with nonnegative coefficients (or alternatively, the unique minimal non trivial annihilating linear combination of the columns of the Cartan matrix with non-negative coefficients).

Throw an error if the existence or uniqueness does not hold

FIXME: the current implementation assumes that the Cartan matrix is indexed by \([0,1,...]\), in the same order as the index set.

EXAMPLES:

sage: # needs sage.graphs
sage: RootSystem(['C',2,1]).cartan_type().a()
Finite family {0: 1, 1: 2, 2: 1}
sage: RootSystem(['D',4,1]).cartan_type().a()
Finite family {0: 1, 1: 1, 2: 2, 3: 1, 4: 1}
sage: RootSystem(['F',4,1]).cartan_type().a()
Finite family {0: 1, 1: 2, 2: 3, 3: 4, 4: 2}
sage: RootSystem(['BC',4,2]).cartan_type().a()
Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1}

a is a shortcut for col_annihilator:

sage: RootSystem(['BC',4,2]).cartan_type().col_annihilator()                # needs sage.graphs
Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1}

acheck(m=None)#

Return the unique minimal non trivial annihilating linear combination of \(\alpha_0, \alpha_1, \ldots, \alpha_n\) with nonnegative coefficients (or alternatively, the unique minimal non trivial annihilating linear combination of the rows of the Cartan matrix with non-negative coefficients).

Throw an error if the existence of uniqueness does not hold

The optional argument m is for internal use only.

EXAMPLES:

sage: # needs sage.graphs
sage: RootSystem(['C',2,1]).cartan_type().acheck()
Finite family {0: 1, 1: 1, 2: 1}
sage: RootSystem(['D',4,1]).cartan_type().acheck()
Finite family {0: 1, 1: 1, 2: 2, 3: 1, 4: 1}
sage: RootSystem(['F',4,1]).cartan_type().acheck()
Finite family {0: 1, 1: 2, 2: 3, 3: 2, 4: 1}
sage: RootSystem(['BC',4,2]).cartan_type().acheck()
Finite family {0: 1, 1: 2, 2: 2, 3: 2, 4: 2}

acheck is a shortcut for row_annihilator:

sage: RootSystem(['BC',4,2]).cartan_type().row_annihilator()                # needs sage.graphs
Finite family {0: 1, 1: 2, 2: 2, 3: 2, 4: 2}

FIXME:

The current implementation assumes that the Cartan matrix is indexed by \([0,1,...]\), in the same order as the index set.
This really should be a method of CartanMatrix.

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: CartanType(['A', 1, 1]).basic_untwisted()
['A', 1]
sage: CartanType(['A', 3, 1]).basic_untwisted()
['A', 3]
sage: CartanType(['B', 3, 1]).basic_untwisted()
['B', 3]
sage: CartanType(['E', 6, 1]).basic_untwisted()
['E', 6]
sage: CartanType(['G', 2, 1]).basic_untwisted()
['G', 2]

sage: CartanType(['A', 2, 2]).basic_untwisted()
['A', 2]
sage: CartanType(['A', 4, 2]).basic_untwisted()
['A', 4]
sage: CartanType(['A', 11, 2]).basic_untwisted()
['A', 11]
sage: CartanType(['D', 5, 2]).basic_untwisted()
['D', 5]
sage: CartanType(['E', 6, 2]).basic_untwisted()
['E', 6]
sage: CartanType(['D', 4, 3]).basic_untwisted()
['D', 4]

c()#

Returns the family (c_i)_i of integer coefficients defined by \(c_i=max(1, a_i/a^vee_i)\) (see e.g. [FSS07] p. 3)

FIXME: the current implementation assumes that the Cartan matrix is indexed by \([0,1,...]\), in the same order as the index set.

EXAMPLES:

sage: # needs sage.graphs
sage: RootSystem(['C',2,1]).cartan_type().c()
Finite family {0: 1, 1: 2, 2: 1}
sage: RootSystem(['D',4,1]).cartan_type().c()
Finite family {0: 1, 1: 1, 2: 1, 3: 1, 4: 1}
sage: RootSystem(['F',4,1]).cartan_type().c()
Finite family {0: 1, 1: 1, 2: 1, 3: 2, 4: 2}
sage: RootSystem(['BC',4,2]).cartan_type().c()
Finite family {0: 2, 1: 1, 2: 1, 3: 1, 4: 1}

REFERENCES:

[FSS07]

G. Fourier, A. Schilling, and M. Shimozono, Demazure structure inside Kirillov-Reshetikhin crystals, J. Algebra, Vol. 309, (2007), p. 386-404 arXiv math/0605451

classical()#

Return the classical Cartan type associated with this affine Cartan type.

EXAMPLES:

sage: CartanType(['A', 1, 1]).classical()
['A', 1]
sage: CartanType(['A', 3, 1]).classical()
['A', 3]
sage: CartanType(['B', 3, 1]).classical()
['B', 3]

sage: CartanType(['A', 2, 2]).classical()
['C', 1]
sage: CartanType(['BC', 1, 2]).classical()
['C', 1]
sage: CartanType(['A', 4, 2]).classical()
['C', 2]
sage: CartanType(['BC', 2, 2]).classical()
['C', 2]
sage: CartanType(['A', 10, 2]).classical()
['C', 5]
sage: CartanType(['BC', 5, 2]).classical()
['C', 5]

sage: CartanType(['D', 5, 2]).classical()
['B', 4]
sage: CartanType(['E', 6, 1]).classical()
['E', 6]
sage: CartanType(['G', 2, 1]).classical()
['G', 2]
sage: CartanType(['E', 6, 2]).classical()
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1}
sage: CartanType(['D', 4, 3]).classical()
['G', 2]

We check that classical(), sage.combinat.root_system.cartan_type.CartanType_crystallographic.dynkin_diagram(), and special_node() are consistent:

sage: for ct in CartanType.samples(affine=True):                            # needs sage.graphs
....:     g1 = ct.classical().dynkin_diagram()
....:     g2 = ct.dynkin_diagram()
....:     g2.delete_vertex(ct.special_node())
....:     assert g1.vertices(sort=True) == g2.vertices(sort=True)
....:     assert g1.edges(sort=True) == g2.edges(sort=True)

col_annihilator()#

Throw an error if the existence or uniqueness does not hold

FIXME: the current implementation assumes that the Cartan matrix is indexed by \([0,1,...]\), in the same order as the index set.

EXAMPLES:

sage: # needs sage.graphs
sage: RootSystem(['C',2,1]).cartan_type().a()
Finite family {0: 1, 1: 2, 2: 1}
sage: RootSystem(['D',4,1]).cartan_type().a()
Finite family {0: 1, 1: 1, 2: 2, 3: 1, 4: 1}
sage: RootSystem(['F',4,1]).cartan_type().a()
Finite family {0: 1, 1: 2, 2: 3, 3: 4, 4: 2}
sage: RootSystem(['BC',4,2]).cartan_type().a()
Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1}

a is a shortcut for col_annihilator:

sage: RootSystem(['BC',4,2]).cartan_type().col_annihilator()                # needs sage.graphs
Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1}

is_affine()#

EXAMPLES:

sage: CartanType(['A', 3, 1]).is_affine()
True

is_finite()#

EXAMPLES:

sage: CartanType(['A', 3, 1]).is_finite()
False

is_untwisted_affine()#

Return whether self is untwisted affine

A Cartan type is untwisted affine if it is the canonical affine extension of some finite type. Every affine type is either untwisted affine, dual thereof, or of type BC.

EXAMPLES:

sage: CartanType(['A', 3, 1]).is_untwisted_affine()
True
sage: CartanType(['A', 3, 1]).dual().is_untwisted_affine()  # this one is self dual!
True
sage: CartanType(['B', 3, 1]).dual().is_untwisted_affine()
False
sage: CartanType(['BC', 3, 2]).is_untwisted_affine()
False

other_affinization()#

Return the other affinization of the same classical type.

EXAMPLES:

sage: CartanType(["A", 3, 1]).other_affinization()
['A', 3, 1]
sage: CartanType(["B", 3, 1]).other_affinization()
['C', 3, 1]^*
sage: CartanType(["C", 3, 1]).dual().other_affinization()
['B', 3, 1]

Is this what we want?:

sage: CartanType(["BC", 3, 2]).dual().other_affinization()
['B', 3, 1]

row_annihilator(m=None)#

Throw an error if the existence of uniqueness does not hold

The optional argument m is for internal use only.

EXAMPLES:

sage: # needs sage.graphs
sage: RootSystem(['C',2,1]).cartan_type().acheck()
Finite family {0: 1, 1: 1, 2: 1}
sage: RootSystem(['D',4,1]).cartan_type().acheck()
Finite family {0: 1, 1: 1, 2: 2, 3: 1, 4: 1}
sage: RootSystem(['F',4,1]).cartan_type().acheck()
Finite family {0: 1, 1: 2, 2: 3, 3: 2, 4: 1}
sage: RootSystem(['BC',4,2]).cartan_type().acheck()
Finite family {0: 1, 1: 2, 2: 2, 3: 2, 4: 2}

acheck is a shortcut for row_annihilator:

sage: RootSystem(['BC',4,2]).cartan_type().row_annihilator()                # needs sage.graphs
Finite family {0: 1, 1: 2, 2: 2, 3: 2, 4: 2}

FIXME:

The current implementation assumes that the Cartan matrix is indexed by \([0,1,...]\), in the same order as the index set.
This really should be a method of CartanMatrix.

special_node()#

Return a special node of the Dynkin diagram.

A special node is a node of the Dynkin diagram such that pruning it yields a Dynkin diagram for the associated classical type (see classical()).

This method returns the label of some special node. This is usually \(0\) in the standard conventions.

EXAMPLES:

sage: CartanType(['A', 3, 1]).special_node()
0

The choice is guaranteed to be consistent with the indexing of the nodes of the classical Dynkin diagram:

sage: CartanType(['A', 3, 1]).index_set()
(0, 1, 2, 3)
sage: CartanType(['A', 3, 1]).classical().index_set()
(1, 2, 3)

special_nodes()#

Return the set of special nodes of the affine Dynkin diagram.

EXAMPLES:

sage: # needs sage.graphs sage.groups
sage: CartanType(['A',3,1]).special_nodes()
(0, 1, 2, 3)
sage: CartanType(['C',2,1]).special_nodes()
(0, 2)
sage: CartanType(['D',4,1]).special_nodes()
(0, 1, 3, 4)
sage: CartanType(['E',6,1]).special_nodes()
(0, 1, 6)
sage: CartanType(['D',3,2]).special_nodes()
(0, 2)
sage: CartanType(['A',4,2]).special_nodes()
(0,)

translation_factors()#

Return the translation factors for self.

Those are the smallest factors \(t_i\) such that the translation by \(t_i \alpha_i\) maps the fundamental polygon to another polygon in the alcove picture.

OUTPUT:

a dictionary from self.index_set() to \(\ZZ\) (or \(\QQ\) for affine type \(BC\))

Those coefficients are all \(1\) for dual untwisted, and in particular for simply laced. They coincide with the usual \(c_i\) coefficients (see c()) for untwisted and dual thereof. See the discussion below for affine type \(BC\).

Note

One usually realizes the alcove picture in the coweight lattice, with translations by coroots; in that case, one will use the translation factors for the dual Cartan type.

FIXME: the current implementation assumes that the Cartan matrix is indexed by \([0,1,...]\), in the same order as the index set.

EXAMPLES:

sage: # needs sage.graphs
sage: CartanType(['C',2,1]).translation_factors()
Finite family {0: 1, 1: 2, 2: 1}
sage: CartanType(['C',2,1]).dual().translation_factors()
Finite family {0: 1, 1: 1, 2: 1}
sage: CartanType(['D',4,1]).translation_factors()
Finite family {0: 1, 1: 1, 2: 1, 3: 1, 4: 1}
sage: CartanType(['F',4,1]).translation_factors()
Finite family {0: 1, 1: 1, 2: 1, 3: 2, 4: 2}
sage: CartanType(['BC',4,2]).translation_factors()
Finite family {0: 1, 1: 1, 2: 1, 3: 1, 4: 1/2}

We proceed with systematic tests taken from MuPAD-Combinat’s testsuite:

sage: # needs sage.graphs
sage: list(CartanType(["A", 1, 1]).translation_factors())
[1, 1]
sage: list(CartanType(["A", 5, 1]).translation_factors())
[1, 1, 1, 1, 1, 1]
sage: list(CartanType(["B", 5, 1]).translation_factors())
[1, 1, 1, 1, 1, 2]
sage: list(CartanType(["C", 5, 1]).translation_factors())
[1, 2, 2, 2, 2, 1]
sage: list(CartanType(["D", 5, 1]).translation_factors())
[1, 1, 1, 1, 1, 1]
sage: list(CartanType(["E", 6, 1]).translation_factors())
[1, 1, 1, 1, 1, 1, 1]
sage: list(CartanType(["E", 7, 1]).translation_factors())
[1, 1, 1, 1, 1, 1, 1, 1]
sage: list(CartanType(["E", 8, 1]).translation_factors())
[1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: list(CartanType(["F", 4, 1]).translation_factors())
[1, 1, 1, 2, 2]
sage: list(CartanType(["G", 2, 1]).translation_factors())
[1, 3, 1]
sage: list(CartanType(["A", 2, 2]).translation_factors())
[1, 1/2]
sage: list(CartanType(["A", 2, 2]).dual().translation_factors())
[1/2, 1]
sage: list(CartanType(["A", 10, 2]).translation_factors())
[1, 1, 1, 1, 1, 1/2]
sage: list(CartanType(["A", 10, 2]).dual().translation_factors())
[1/2, 1, 1, 1, 1, 1]
sage: list(CartanType(["A", 9, 2]).translation_factors())
[1, 1, 1, 1, 1, 1]
sage: list(CartanType(["D", 5, 2]).translation_factors())
[1, 1, 1, 1, 1]
sage: list(CartanType(["D", 4, 3]).translation_factors())
[1, 1, 1]
sage: list(CartanType(["E", 6, 2]).translation_factors())
[1, 1, 1, 1, 1]

We conclude with a discussion of the appropriate value for affine type \(BC\). Let us consider the alcove picture realized in the weight lattice. It is obtained by taking the level-\(1\) affine hyperplane in the weight lattice, and projecting it along \(\Lambda_0\):

sage: R = RootSystem(["BC",2,2])
sage: alpha = R.weight_space().simple_roots()                               # needs sage.graphs
sage: alphacheck = R.coroot_space().simple_roots()
sage: Lambda = R.weight_space().fundamental_weights()

Here are the levels of the fundamental weights:

sage: Lambda[0].level(), Lambda[1].level(), Lambda[2].level()               # needs sage.graphs
(1, 2, 2)

So the “center” of the fundamental polygon at level \(1\) is:

sage: O = Lambda[0]
sage: O.level()                                                             # needs sage.graphs
1

We take the projection \(\omega_1\) at level \(0\) of \(\Lambda_1\) as unit vector on the \(x\)-axis, and the projection \(\omega_2\) at level 0 of \(\Lambda_2\) as unit vector of the \(y\)-axis:

sage: omega1 = Lambda[1] - 2*Lambda[0]
sage: omega2 = Lambda[2] - 2*Lambda[0]
sage: omega1.level(), omega2.level()                                        # needs sage.graphs
(0, 0)

The projections of the simple roots can be read off:

sage: alpha[0]                                                              # needs sage.graphs
2*Lambda[0] - Lambda[1]
sage: alpha[1]                                                              # needs sage.graphs
-2*Lambda[0] + 2*Lambda[1] - Lambda[2]
sage: alpha[2]                                                              # needs sage.graphs
-2*Lambda[1] + 2*Lambda[2]

Namely \(\alpha_0 = -\omega_1\), \(\alpha_1 = 2\omega_1 - \omega_2\) and \(\alpha_2 = -2 \omega_1 + 2 \omega_2\).

The reflection hyperplane defined by \(\alpha_0^\vee\) goes through the points \(O+1/2 \omega_1\) and \(O+1/2 \omega_2\):

sage: (O+(1/2)*omega1).scalar(alphacheck[0])
0
sage: (O+(1/2)*omega2).scalar(alphacheck[0])
0

Hence, the fundamental alcove is the triangle \((O, O+1/2 \omega_1, O+1/2 \omega_2)\). By successive reflections, one can tile the full plane. This induces a tiling of the full plane by translates of the fundamental polygon.

Todo

Add the picture here, once root system plots in the weight lattice will be implemented. In the mean time, the reader may look up the dual picture on Figure 2 of [HST09] which was produced by MuPAD-Combinat.

From this picture, one can read that translations by \(\alpha_0\), \(\alpha_1\), and \(1/2\alpha_2\) map the fundamental polygon to translates of it in the alcove picture, and are smallest with this property. Hence, the translation factors for affine type \(BC\) are \(t_0=1, t_1=1, t_2=1/2\):

sage: CartanType(['BC',2,2]).translation_factors()                          # needs sage.graphs
Finite family {0: 1, 1: 1, 2: 1/2}

REFERENCES:

[HST09]

F. Hivert, A. Schilling, and N. M. Thiery, Hecke group algebras as quotients of affine Hecke algebras at level 0, JCT A, Vol. 116, (2009) p. 844-863 arXiv 0804.3781

class sage.combinat.root_system.cartan_type.CartanType_crystallographic#

Bases: CartanType_abstract

An abstract class for crystallographic Cartan types.

ascii_art(label='lambda x: x', node=None)#

Return an ascii art representation of the Dynkin diagram.

INPUT:

label – (default: the identity) a relabeling function for the nodes
node – (optional) a function which returns the character for a node

EXAMPLES:

sage: cartan_type = CartanType(['B',5,1])
sage: print(cartan_type.ascii_art())
    O 0
    |
    |
O---O---O---O=>=O
1   2   3   4   5

The label option is useful to visualize various statistics on the nodes of the Dynkin diagram:

sage: a = cartan_type.col_annihilator(); a                                  # needs sage.graphs
Finite family {0: 1, 1: 1, 2: 2, 3: 2, 4: 2, 5: 2}
sage: print(CartanType(['B',5,1]).ascii_art(label=a.__getitem__))           # needs sage.graphs
    O 1
    |
    |
O---O---O---O=>=O
1   2   2   2   2

cartan_matrix()#

Return the Cartan matrix associated with self.

EXAMPLES:

sage: CartanType(['A',4]).cartan_matrix()                                   # needs sage.graphs
[ 2 -1  0  0]
[-1  2 -1  0]
[ 0 -1  2 -1]
[ 0  0 -1  2]

coxeter_diagram()#

Return the Coxeter diagram for self.

This implementation constructs it from the Dynkin diagram.