Cartan types¶

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.

http://en.wikipedia.org/wiki/Dynkin_diagram

Let us consider for example, the Cartan type $A_4$ :

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

It is the name of the following Dynkin diagram:

sage: DynkinDiagram(T)
O---O---O---O
1   2   3   4
A4

Note: for convenience, the following shortcuts are available:

sage: DynkinDiagram(['A',4])
O---O---O---O
1   2   3   4
A4
sage: DynkinDiagram('A4')
O---O---O---O
1   2   3   4
A4
sage: T.dynkin_diagram()
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 is the symmetric group $S_{n+1}$ :

sage: W = WeylGroup(T)
sage: W
Weyl Group of type ['A', 4] (as a matrix group acting on the ambient space)
sage: W.cardinality()
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 = CrystalOfLetters(T)
sage: C
The crystal of letters for type ['A', 4]
sage: C.list()
[1, 2, 3, 4, 5]

sage: C = CrystalOfTableaux(T, shape=[2])
sage: C
The crystal of tableaux of type ['A', 4] and shape(s) [[2]]
sage: C.cardinality()
15

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

sage: CartanType.samples(finite = True, crystalographic = 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]]

Non-crystallographic Cartan types are also partially supported:

sage: CartanType.samples(finite = True, crystalographic = 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 http://en.wikipedia.org/wiki/Root_system), and further mathematical properties:

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

It will eventually include Coxeter numbers, etc.

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)
    O 4
    |
    |
O---O---O
1   2   3
D4

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

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()
        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 feature 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()
sage: g.add_vertices([1,2,3])

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

sage: g.cartan_matrix()
[2 0 0]
[0 2 0]
[0 0 2]

Then we add a couple edges:

sage: g.add_edge(1,2,2)
sage: g.add_edge(1,3)
sage: g.add_edge(2,3)

and we get the desired Cartan matrix:

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

Note that backward edges have been automatically added:

sage: g.edges()
[(1, 2, 2), (1, 3, 1), (2, 1, 1), (2, 3, 1), (3, 1, 1), (3, 2, 1)]

Caveat: the Dynkin diagram should not be modified after having been used; this is not checked currently (Todo: add a method set_mutable(), as for matrices).

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

or dual thereof:

sage: CartanType(["B", 4, 1]).dual().dynkin_diagram()
    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()
O=<=O---O---O=<=O
0   1   2   3   4
BC4~

This includes the two degenerate cases:

sage: CartanType(["A", 1, 1]).dynkin_diagram()
O<=>O
0   1
A1~
sage: CartanType(["BC", 1, 2]).dynkin_diagram()
  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: 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~*

class sage.combinat.root_system.cartan_type.CartanTypeFactory¶

Bases: sage.structure.sage_object.SageObject

samples(finite=None, affine=None, crystalographic=None)¶

Returns a sample of the available Cartan types.

INPUT:

finite: a boolean or None; defaults to None.
affine: a boolean or None; defaults to None.
crystalographic: a boolean or None; defaults to 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],
 ['B', 5, 1]^*, ['C', 4, 1]^*, ['F', 4, 1]^*, ['G', 2, 1]^*,
 ['BC', 1, 2], ['BC', 5, 2]]

The finite, affine and crystalographic options allow respectively for restricting to (non) finite, (non) affine, and (non) crystalographic 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],
 ['B', 5, 1]^*, ['C', 4, 1]^*, ['F', 4, 1]^*, ['G', 2, 1]^*,
 ['BC', 1, 2], ['BC', 5, 2]]

sage: CartanType.samples(crystalographic=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],
 ['B', 5, 1]^*, ['C', 4, 1]^*, ['F', 4, 1]^*, ['G', 2, 1]^*,
 ['BC', 1, 2], ['BC', 5, 2]]

sage: CartanType.samples(crystalographic=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()

coxeter_diagram()¶: Returns the Coxeter diagram for self.

dual()¶

Returns 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]^*

index_set()¶

Returns the index set for self. This is the list of the nodes of the associated Coxeter / 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()¶

Returns 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 :meth:’.is_reducible’, except for the Cartan type $D_2$ .

$D_2$ is not a standard Cartan type. It is equivalent to type $A_1 imes A_1$ which is reducible; however the isomorphism from its ambient space (for the orthogonal group of degree 4) to that of $A_1 imes A_1$ 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

TESTS:

sage: all( T.is_irreducible() == T.is_atomic() for T in CartanType.samples() if T != CartanType("D2"))
True

is_compound()¶

A short hand for not is_atomic().

TESTS:

sage: all( T.is_compound() == (not T.is_atomic()) for T in CartanType.samples())
True

is_crystalographic()¶

Returns whether this Cartan type is crystalographic.

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

EXAMPLES:

sage: [ [t, t.is_crystalographic() ] 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]]

TESTS:

sage: all(t.is_crystalographic() for t in CartanType.samples(affine=True))
True

is_finite()¶

Returns 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()
...
NotImplementedError: <abstract method is_finite at ...>

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

is_implemented()¶

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()¶

Returns 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],
 [['B', 5, 1]^*, False], [['C', 4, 1]^*, False], [['F', 4, 1]^*, False], [['G', 2, 1]^*, False],
 [['BC', 1, 2], False], [['BC', 5, 2], False]]

rank()¶

Returns the rank of self, that is the number of nodes of the Dynkin diagram

EXAMPLES:

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

relabel(relabelling)¶

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:

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

root_system()¶

Returns the root system associated to self.

EXAMPLES:

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

type()¶

Returns 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: sage.combinat.root_system.cartan_type.CartanType_simple, sage.combinat.root_system.cartan_type.CartanType_crystalographic

An abstract class for simple affine Cartan types

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: 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}

acheck(m=None)¶

Return the unique minimal non trivial annihilating linear combination of $\alpha_0, \alpha1, \ldots, \alpha_n$ 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: 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}

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

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: 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}

TESTS:

sage: CartanType(["B", 3, 1]).c().map(parent)
Finite family {0: Integer Ring, 1: Integer Ring, 2: Integer Ring, 3: Integer Ring}

REFERENCES:

[FSS07]

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

classical()¶

Returns 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() # todo: double check
['F', 4]^*
sage: CartanType(['D', 4, 3]).classical() # todo: double check
['G', 2]^* relabelled by {1: 2, 2: 1}

We check that classical(), dynkin_diagram(), and special_node() are consistent:

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

col_annihilator()¶

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: 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}

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()¶

Returns whether self is untwisted affine, that is if it is the affine extension of some finite type.

Every affine type is either untwisted affine, or 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

row_annihilator(m=None)¶

Return the unique minimal non trivial annihilating linear combination of $\alpha_0, \alpha1, \ldots, \alpha_n$ 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: 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}

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()¶

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]

translation_factors()¶

Returns 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: 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: 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()
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()
(1, 2, 2)

So the “center” of the fundamental polygon at level $1$ is:

sage: O = Lambda[0]
sage: O.level()
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()
(0, 0)

The projections of the simple roots can be read off:

sage: alpha[0]
2*Lambda[0] - Lambda[1]
sage: alpha[1]
-2*Lambda[0] + 2*Lambda[1] - Lambda[2]
sage: alpha[2]
-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()
Finite family {0: 1, 1: 1, 2: 1/2}

TESTS:

sage: CartanType(["B", 3, 1]).translation_factors().map(parent)
Finite family {0: Integer Ring, 1: Integer Ring, 2: Integer Ring, 3: Integer Ring}
sage: CartanType(["BC", 3, 2]).translation_factors().map(parent)
Finite family {0: Integer Ring, 1: Integer Ring, 2: Integer Ring, 3: Rational Field}

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 http://arxiv.org/abs/0804.3781

class sage.combinat.root_system.cartan_type.CartanType_crystalographic¶

Bases: sage.combinat.root_system.cartan_type.CartanType_abstract

An abstract class for crystalographic cartan types

cartan_matrix()¶

Returns the Cartan matrix associated with self.

EXAMPLES:

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

dynkin_diagram()¶

Returns the Dynkin diagram associated with self.

EXAMPLES:

sage: CartanType(['A',4]).dynkin_diagram()
O---O---O---O
1   2   3   4
A4

Note: derived subclasses should typically implement this as a cached method

is_crystalographic()¶

Implements CartanType_abstract.is_crystalographic() by returning $True$ .

EXAMPLES:

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

symmetrizer()¶

A Cartan matrix $M$ is symmetrizable if there exists a non trivial diagonal matrix $D$ such that $DM$ is a symmetric matrix, that is $DM = M^tD$ . In that case, $D$ is unique, up to a scalar factor for each connected component of the Dynkin diagram.

This method currently assumes that the Cartan type is irreducible.

This method computes the unique minimal such $D$ with non negative integral coefficients. If $D$ exists, it is returned as a family. Otherwise None is returned.

EXAMPLES:

sage: CartanType(["B",5]).symmetrizer()
Finite family {1: 2, 2: 2, 3: 2, 4: 2, 5: 1}

Here is a neat trick to visualize it better:

sage: T = CartanType(["B",5])
sage: print T.ascii_art(T.symmetrizer().__getitem__)
O---O---O---O=>=O
2   2   2   2   1

sage: T = CartanType(["BC",5, 2])
sage: print T.ascii_art(T.symmetrizer().__getitem__)
O=<=O---O---O---O=<=O
1   2   2   2   2   4

Property: up to an overall scalar factor, this gives the norm of the simple roots in the ambient space:

sage: T = CartanType(["C",5])
sage: print T.ascii_art(T.symmetrizer().__getitem__)
O---O---O---O=<=O
1   1   1   1   2

sage: alpha = RootSystem(T).ambient_space().simple_roots()
sage: print T.ascii_art(lambda i: alpha[i].scalar(alpha[i]))
O---O---O---O=<=O
2   2   2   2   4

class sage.combinat.root_system.cartan_type.CartanType_finite¶

Bases: sage.combinat.root_system.cartan_type.CartanType_abstract

An abstract class for simple affine Cartan types

is_affine()¶

EXAMPLES:

sage: CartanType(["A", 3]).is_affine()
False

is_finite()¶

EXAMPLES:

sage: CartanType(["A", 3]).is_finite()
True

class sage.combinat.root_system.cartan_type.CartanType_simple¶

Bases: sage.combinat.root_system.cartan_type.CartanType_abstract

An abstract class for simple Cartan types

is_irreducible()¶

EXAMPLES:

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

class sage.combinat.root_system.cartan_type.CartanType_simple_finite¶: Bases: object

class sage.combinat.root_system.cartan_type.CartanType_simply_laced¶

Bases: sage.combinat.root_system.cartan_type.CartanType_crystalographic

An abstract class for simply laced cartan types

dual()¶

Simply laced cartan types are self dual

EXAMPLES:

sage: CartanType(["A", 3]).dual()
['A', 3]
sage: CartanType(["A", 3, 1]).dual()
['A', 3, 1]
sage: CartanType(["D", 3]).dual()
['D', 3]
sage: CartanType(["D", 4, 1]).dual()
['D', 4, 1]
sage: CartanType(["E", 6]).dual()
['E', 6]
sage: CartanType(["E", 6, 1]).dual()
['E', 6, 1]

is_simply_laced()¶

EXAMPLES:

sage: CartanType(['A',3,1]).is_simply_laced()
True
sage: CartanType(['A',2]).is_simply_laced()
True

class sage.combinat.root_system.cartan_type.CartanType_standard_affine(letter, n, affine=1)¶

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

A concrete class for affine simple Cartan types

index_set()¶

Implements CartanType_abstract.index_set().

The index set for all standard affine Cartan types is of the form ${0,\dots,n}$ .

EXAMPLES::: sage: CartanType([‘A’, 5, 1]).index_set() [0, 1, 2, 3, 4, 5]

rank()¶

EXAMPLES:

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

special_node()¶

Implements :meth:’CartanType_abstract.special_node`.

With the standard labelling conventions, $0$ is always a special node.

EXAMPLES:

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

type()¶

Returns the type of self.

EXAMPLES:

sage: CartanType(['A', 4, 1]).type()
'A'

class sage.combinat.root_system.cartan_type.CartanType_standard_finite(letter, n)¶

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

A concrete base class for the finite standard Cartan types $A_3$ , $D_4$ , $E_8$

affine()¶

Returns the corresponding untwisted affine Cartan type

EXAMPLES:

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

index_set()¶

Implements CartanType_abstract.index_set().

The index set for all standard finite Cartan types is of the form ${1,\dots,n}$ . (but see type_I for a slight abuse of this).

EXAMPLES::: sage: CartanType([‘A’, 5]).index_set() [1, 2, 3, 4, 5]

rank()¶

EXAMPLES:

sage: CartanType(['A', 3]).rank()
3
sage: CartanType(['B', 3]).rank()
3
sage: CartanType(['C', 3]).rank()
3
sage: CartanType(['D', 4]).rank()
4
sage: CartanType(['E', 6]).rank()
6

type()¶

Returns the type of self.

EXAMPLES:

sage: CartanType(['A', 4]).type()
'A'
sage: CartanType(['A', 4, 1]).type()
'A'

class sage.combinat.root_system.cartan_type.CartanType_standard_untwisted_affine(letter, n, affine=1)¶

Bases: sage.combinat.root_system.cartan_type.CartanType_standard_affine

A concrete class for the standard untwisted affine Cartan types

classical()¶

Returns the classical Cartan type associated with self (which should be affine)

EXAMPLES:

sage: CartanType(['A', 3, 1]).classical()
['A', 3]
sage: CartanType(['B', 3, 1]).classical()
['B', 3]
sage: CartanType(['C', 3, 1]).classical()
['C', 3]
sage: CartanType(['D', 4, 1]).classical()
['D', 4]
sage: CartanType(['E', 6, 1]).classical()
['E', 6]
sage: CartanType(['F', 4, 1]).classical()
['F', 4]
sage: CartanType(['G', 2, 1]).classical()
['G', 2]

is_untwisted_affine()¶

Implements :meth:’CartanType_affine.is_untwisted_affine` by returning True.

EXAMPLES:

sage: CartanType(['B', 3, 1]).is_untwisted_affine()
True

Cartan types¶

Previous topic

Next topic

This Page

Navigation

Cartan types¶

Previous topic

Next topic

This Page

Quick search

Navigation