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# -*- coding: utf-8 -*-

r"""

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:`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

.. NOTE::

For convenience, the following shortcuts are available::

sage: DynkinDiagram(['A',4])

O---O---O---O

1 2 3 4

sage: DynkinDiagram('A4')

O---O---O---O

1 2 3 4

sage: T.dynkin_diagram()

O---O---O---O

1 2 3 4

See :class:`~sage.combinat.root_system.dynkin_diagram.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)

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 = crystals.Letters(T)

sage: C

The crystal of letters for type ['A', 4]

sage: C.list()

[1, 2, 3, 4, 5]

sage: C = crystals.Tableaux(T, shape=[2])

sage: C

The crystal of tableaux of type ['A', 4] and shape(s) [[2]]

sage: C.cardinality()

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. :mod:`sage.combinat.root_system.type_A`).

This database includes how to construct the Dynkin diagram, the ambient space

for the root system (see :wikipedia:`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)

O 4

O---O---O

1 2 3

sage: E6 = CartanType(['E',6])

sage: DynkinDiagram(E6)

O 2

O---O---O---O---O

1 3 4 5 6

.. 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

O=<=O

1 2

sage: C2.cartan_matrix()

[ 2 -2]

[-1 2]

However Bourbaki would have the Cartan matrix as:

.. MATH::

\begin{bmatrix}

2 & -1 \\

-2 & 2

\end{bmatrix}.

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 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()

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 0 0]

[0 2 0]

[0 0 2]

Oops, the Cartan matrix did not change! This is because it is cached

for efficiency (see :class:`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 :meth:`set_mutable` as, say, for matrices

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

sage: delattr(g, 'cartan_matrix')

Now 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)]

.. rubric:: 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

.. rubric:: 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]

.. rubric:: 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()

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()

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()

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: 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()

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'

.. rubric:: 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]

.. rubric:: Abstract classes for Cartan types

- :class:`CartanType_abstract`

- :class:`CartanType_crystallographic`

- :class:`CartanType_simply_laced`

- :class:`CartanType_simple`

- :class:`CartanType_finite`

- :class:`CartanType_affine` (see also :ref:`sage.combinat.root_system.type_affine`)

- :obj:`sage.combinat.root_system.cartan_type.CartanType`

- :ref:`sage.combinat.root_system.type_dual`

- :ref:`sage.combinat.root_system.type_reducible`

- :ref:`sage.combinat.root_system.type_relabel`

Concrete classes for Cartan types

- :class:`CartanType_standard`

- :class:`CartanType_standard_finite`

- :class:`CartanType_standard_affine`

- :class:`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.

- :ref:`sage.combinat.root_system.type_A`

- :ref:`sage.combinat.root_system.type_B`

- :ref:`sage.combinat.root_system.type_C`

- :ref:`sage.combinat.root_system.type_D`

- :ref:`sage.combinat.root_system.type_E`

- :ref:`sage.combinat.root_system.type_F`

- :ref:`sage.combinat.root_system.type_G`

- :ref:`sage.combinat.root_system.type_H`

- :ref:`sage.combinat.root_system.type_I`

- :ref:`sage.combinat.root_system.type_A_affine`

- :ref:`sage.combinat.root_system.type_B_affine`

- :ref:`sage.combinat.root_system.type_C_affine`

- :ref:`sage.combinat.root_system.type_D_affine`

- :ref:`sage.combinat.root_system.type_E_affine`

- :ref:`sage.combinat.root_system.type_F_affine`

- :ref:`sage.combinat.root_system.type_G_affine`

- :ref:`sage.combinat.root_system.type_BC_affine`

- :ref:`sage.combinat.root_system.type_A_infinity`

.. TODO:: Should those indexes come before the introduction?

"""

#*****************************************************************************

# This program is free software: you can redistribute it and/or modify

# it under the terms of the GNU General Public License as published by

# the Free Software Foundation, either version 2 of the License, or

# (at your option) any later version.

# http://www.gnu.org/licenses/

#*****************************************************************************

from __future__ import print_function, absolute_import, division

from six.moves import range

from six.moves.builtins import sorted

from six import class_types, string_types

from sage.misc.cachefunc import cached_method

from sage.misc.abstract_method import abstract_method

from sage.misc.lazy_import import LazyImport

from sage.rings.all import ZZ

from sage.rings.infinity import Infinity

from sage.structure.sage_object import SageObject

from sage.structure.unique_representation import UniqueRepresentation

from sage.structure.global_options import GlobalOptions

from sage.sets.family import Family

# TODO:

# Implement the Kac conventions by relabeling/dual/... of the above

# Implement Coxeter diagrams for non crystallographic

# Intention: we want simultaneously CartanType to be a factory for

# the various subtypes of CartanType_abstract, as in:

# CartanType(["A",4,1])

# and to behaves as a "module" for some extra utilities:

# CartanType.samples()

# Implementation: CartanType is the unique instance of this class

# CartanTypeFactory. Is there a better/more standard way to do it?

class CartanTypeFactory(SageObject):

def __call__(self, *args):

"""

Constructs a Cartan type object.

INPUT:

- ``[letter, rank]`` -- letter is one of 'A', 'B', 'C', 'D', 'E', 'F', 'G'

and rank is an integer or a pair of integers

- ``[letter, rank, twist]`` -- letter is one of 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'BC'

and rank and twist are integers

- ``str`` -- a string

- ``object`` -- a Cartan type, or an object with a Cartan type method

EXAMPLES:

We construct the Cartan type `D_4`::

sage: d4 = CartanType(['D',4])

sage: d4

['D', 4]

or, for short::

sage: CartanType("D4")

['D', 4]

.. SEEALSO:: :func:`~sage.combinat.root_system.cartan_type.CartanType`

TESTS:

Check that this is compatible with :class:`CartanTypeFolded`::

sage: fct = CartanType(['C', 4, 1]).as_folding()

sage: CartanType(fct)

['C', 4, 1]

Check that :trac:`13774` is fixed::

sage: CT = CartanType([['A',2]])

sage: CT.is_irreducible()

True

sage: CT.cartan_matrix()

[ 2 -1]

[-1 2]

sage: CT = CartanType(['A2'])

sage: CT.is_irreducible()

True

sage: CartanType('A2')

['A', 2]

Check that we can pass any Cartan type as a single element list::

sage: CT = CartanType(['A2', 'A2', 'A2'])

sage: CartanType([CT])

A2xA2xA2

sage: CT = CartanType('A2').relabel({1:-1, 2:-2})

sage: CartanType([CT])

['A', 2] relabelled by {1: -1, 2: -2}

Check the errors from :trac:`20973`::

sage: CartanType(['A',-1])

Traceback (most recent call last):

...

ValueError: ['A', -1] is not a valid Cartan type

Check that unicode is handled properly (:trac:`23323`)::

sage: CartanType(u"A3")

['A', 3]

"""

if len(args) == 1:

t = args[0]

else:

t = args

if isinstance(t, (CartanType_abstract, SuperCartanType_standard)):

return t

if hasattr(t, "cartan_type"):

return t.cartan_type()

if len(t) == 1: # Fix for trac #13774

t = t[0]

# We need to make another check

if isinstance(t, (CartanType_abstract, SuperCartanType_standard)):

return t

from sage.rings.semirings.non_negative_integer_semiring import NN

if isinstance(t, string_types):

if "x" in t:

from . import type_reducible

return type_reducible.CartanType([CartanType(u) for u in t.split("x")])

elif t[-1] == "*":

return CartanType(t[:-1]).dual()

elif t[-1] == "~":

return CartanType(t[:-1]).affine()

elif t in ["Aoo", u"A∞"]:

return CartanType(['A', Infinity])

elif t == "A+oo":

from . import type_A_infinity

return type_A_infinity.CartanType(NN)

else:

return CartanType([t[0], eval(t[1:])])

t = list(t)

if isinstance(t[0], string_types) and t[1] in [Infinity, ZZ, NN]:

letter, n = t[0], t[1]

if letter == 'A':

from . import type_A_infinity

if t[1] == NN:

return type_A_infinity.CartanType(NN)

else:

return type_A_infinity.CartanType(ZZ)

if isinstance(t[0], string_types) and t[1] in ZZ and t[1] >= 0:

letter, n = t[0], t[1]

if len(t) == 2:

if letter == "A":

if n >= 0:

from . import type_A

return type_A.CartanType(n)

if letter == "B":

if n >= 1:

from . import type_B

return type_B.CartanType(n)

if letter == "C":

if n >= 1:

from . import type_C

return type_C.CartanType(n)

if letter == "D":

from . import type_D

if n >= 2:

return type_D.CartanType(n)

if letter == "E":

if n >= 6 and n <= 8:

from . import type_E

return type_E.CartanType(n)

if letter == "F":

if n == 4:

from . import type_F

return type_F.CartanType()

if letter == "G":

if n == 2:

from . import type_G

return type_G.CartanType()

if letter == "H":

if n in [3, 4]:

from . import type_H

return type_H.CartanType(n)

if letter == "I":

if n == 1:

return CartanType([["A", 1], ["A", 1]])

if n == 3:

return CartanType(["A", 2])

if n == 4:

return CartanType(["C", 2])

if n == 6:

return CartanType(["G", 2])

if n >= 1:

from . import type_I

return type_I.CartanType(n)

if len(t) == 3:

if t[2] == 1: # Untwisted affine

if letter == "A":

if n >= 1:

from . import type_A_affine

return type_A_affine.CartanType(n)

if letter == "B":

if n >= 1:

from . import type_B_affine

return type_B_affine.CartanType(n)

if letter == "C":

if n >= 1:

from . import type_C_affine

return type_C_affine.CartanType(n)

if letter == "D":

from . import type_D_affine

if n >= 3:

return type_D_affine.CartanType(n)

if letter == "E":

if n >= 6 and n <= 8:

from . import type_E_affine

return type_E_affine.CartanType(n)

if letter == "F":

if n == 4:

from . import type_F_affine

return type_F_affine.CartanType()

if letter == "G":

if n == 2:

from . import type_G_affine

return type_G_affine.CartanType()

if t[2] in [2,3]:

if letter == "BC" and t[2] == 2:

if n >= 1:

from . import type_BC_affine

return type_BC_affine.CartanType(n)

if letter == "A" and t[2] == 2:

if n % 2 == 0: # Kac' A_2n^(2)

return CartanType(["BC", ZZ(n//2), 2])

else: # Kac' A_2n-1^(2)

return CartanType(["B", ZZ((n+1)//2), 1]).dual()

if letter == "D" and t[2] == 2:

return CartanType(["C", n-1, 1]).dual()

if letter == "D" and t[2] == 3 and n == 4:

return CartanType(["G", 2, 1]).dual().relabel([0,2,1])

if letter == "E" and t[2] == 2 and n == 6:

return CartanType(["F", 4, 1]).dual()

raise ValueError("%s is not a valid Cartan type" % t)

if isinstance(t[0], string_types) and isinstance(t[1], (list, tuple)):

letter, n = t[0], t[1]

if len(t) == 2 and len(n) == 2:

from . import type_super_A

return type_super_A.CartanType(n[0], n[1])

raise ValueError("%s is not a valid super Cartan type"%t)

# As the Cartan type has not been recognised try subtypes - but check

# for the error noted in trac:???

from . import type_reducible

try:

return type_reducible.CartanType([ CartanType(subtype) for subtype in t ])

except (SyntaxError, ValueError):

raise ValueError("%s is not a valid Cartan type"%t)

def _repr_(self):

"""

EXAMPLES::

sage: CartanType # indirect doctest

CartanType

"""

return "CartanType"

def samples(self, 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?)

TESTS::

sage: for ct in CartanType.samples(): TestSuite(ct).run()

"""

result = self._samples()

if crystallographic is not None:

result = [t for t in result if t.is_crystallographic() == crystallographic ]

if finite is not None:

result = [t for t in result if t.is_finite() == finite]

if affine is not None:

result = [t for t in result if t.is_affine() == affine]

return result

@cached_method

def _samples(self):

"""

Return a sample of all implemented Cartan types.

.. NOTE:: This is intended to be used through :meth:`samples`.

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]^*]

"""

finite_crystallographic = \

[CartanType (t) for t in [['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]]]

# Support for hand constructed Dynkin diagrams as Cartan types is not yet ready enough for including an example here.

# from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class

# g = DynkinDiagram_class.an_instance()

return finite_crystallographic + \

[CartanType(t) for t in [["I", 5], ["H", 3], ["H", 4]]] + \

[t.affine() for t in finite_crystallographic if t.is_irreducible()] + \

[CartanType(t) for t in [["BC", 1, 2], ["BC", 5, 2]]] + \

[CartanType(t).dual() for t in [["B", 5, 1], ["C", 4, 1], ["F", 4, 1], ["G", 2, 1],["BC", 1, 2], ["BC", 5, 2]]] #+ \

# [ g ]

_colors = {1: 'blue', -1: 'blue',

2: 'red', -2: 'red',

3: 'green', -3: 'green',

4: 'cyan', -4: 'cyan',

5: 'magenta', -5: 'magenta',

6: 'yellow', -6: 'yellow'}

@classmethod

def color(cls, 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'

"""

return cls._colors.get(i, 'black')

# add options to class

class options(GlobalOptions):

r"""

Sets and displays the options for Cartan types. If no parameters

are set, then the function returns a copy of the options dictionary.

The ``options`` to partitions can be accessed as the method

:obj:`CartanType.options` of

:class:`CartanType <CartanTypeFactory>`.

@OPTIONS@

EXAMPLES::

sage: ct = CartanType(['D',5,2]); ct

['C', 4, 1]^*

sage: ct.dynkin_diagram()

O=<=O---O---O=>=O

0 1 2 3 4

C4~*

sage: latex(ct)

C_{4}^{(1)\vee}

sage: CartanType.options(dual_str='#', dual_latex='\\ast',)

sage: ct

['C', 4, 1]^#

sage: ct.dynkin_diagram()

O=<=O---O---O=>=O

0 1 2 3 4

C4~#

sage: latex(ct)

C_{4}^{(1)\ast}

sage: CartanType.options(notation='kac', mark_special_node='both')

sage: ct

['D', 5, 2]

sage: ct.dynkin_diagram()

@=<=O---O---O=>=O

0 1 2 3 4

D5^2

sage: latex(ct)

D_{5}^{(2)}

For type `A_{2n}^{(2)\dagger}`, the dual string/latex options are

automatically overriden::

sage: dct = CartanType(['A',8,2]).dual(); dct

['A', 8, 2]^+

sage: latex(dct)

A_{8}^{(2)\dagger}

sage: dct.dynkin_diagram()

@=>=O---O---O=>=O

0 1 2 3 4

A8^2+

sage: CartanType.options._reset()

"""

NAME = 'CartanType'

module = 'sage.combinat.root_system.cartan_type'

option_class = 'CartanTypeFactory'

notation = dict(default="Stembridge",

description='Specifies which notation Cartan types should use when printed',

values=dict(Stembridge="use Stembridge's notation",

Kac="use Kac's notation"),

case_sensitive=False,

alias=dict(BC="Stembridge", tilde="Stembridge", twisted="Kac"))

dual_str = dict(default="*",

description='The string used for dual Cartan types when printing',

checker=lambda char: isinstance(char, string_types))

dual_latex = dict(default="\\vee",

description='The latex used for dual CartanTypes when latexing',

checker=lambda char: isinstance(char, string_types))

mark_special_node = dict(default="none",

description="Make the special nodes",

values=dict(none="no markup", latex="only in latex",

printing="only in printing", both="both in latex and printing"),

case_sensitive=False)

special_node_str = dict(default="@",

description="The string used to indicate which node is special when printing",

checker=lambda char: isinstance(char, string_types))

marked_node_str = dict(default="X",

description="The string used to indicate a marked node when printing",

checker=lambda char: isinstance(char, string_types))

latex_relabel = dict(default=True,

description="Indicate in the latex output if a Cartan type has been relabelled",

checker=lambda x: isinstance(x, bool))

latex_marked = dict(default=True,

description="Indicate in the latex output if a Cartan type has been marked",

checker=lambda x: isinstance(x, bool))

CartanType = CartanTypeFactory()

CartanType.__doc__ = __doc__

class CartanType_abstract(object):

r"""

Abstract class for Cartan types

Subclasses should implement:

- :meth:`dynkin_diagram()`

- :meth:`cartan_matrix()`

- :meth:`is_finite()`

- :meth:`is_affine()`

- :meth:`is_irreducible()`

"""

def type(self):

r"""

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

"""

return None

def _add_abstract_superclass(self, classes):

"""

Add abstract super-classes to the class of ``self``.

INPUT:

- ``classes`` -- an abstract class or tuple thereof

EXAMPLES::

sage: C = CartanType(["A",3,1])

sage: class MyCartanType:

....: def my_method(self):

....: return 'I am here!'

sage: C._add_abstract_superclass(MyCartanType)

sage: C.__class__

sage: C.__class__.__bases__

(<class 'sage.combinat.root_system.type_A_affine.CartanType_with_superclass'>,

<class __main__.MyCartanType at ...>)

sage: C.my_method()

'I am here!'

.. TODO:: Generalize to :class:`SageObject`?

"""

from sage.structure.dynamic_class import dynamic_class

assert isinstance(classes, (tuple, class_types))

if not isinstance(classes, tuple):

classes = (classes,)

bases = (self.__class__,) + classes

self.__class__ = dynamic_class(self.__class__.__name__+"_with_superclass", bases)

def _ascii_art_node(self, label):

"""

Return the ascii art for the node labelled by ``label``.

EXAMPLES::

sage: CartanType(['A',3])._ascii_art_node(2)

'O'

"""

return "O"

def _latex_draw_node(self, x, y, label, position="below=4pt", fill='white'):

r"""

Draw (possibly marked [crossed out]) circular node ``i`` at the

position ``(x,y)`` with node label ``label`` .

- ``position`` -- position of the label relative to the node

- ``anchor`` -- (optional) the anchor point for the label

EXAMPLES::

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

'\\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$1$};\n'

"""

return "\\draw[fill={}] ({} cm, {} cm) circle (.25cm) node[{}]{{${}$}};\n".format(

fill, x, y, position, label)

def _latex_draw_arrow_tip(self, x, y, rot=0):

r"""

Draw an arrow tip at the point ``(x, y)`` rotated by ``rot``

INPUT:

- ``(x, y)`` -- the coordinates of a point, in cm

- ``rot`` -- an angle, in degrees

This is an internal function used to assist drawing the Dynkin

diagrams. See e.g. :meth:`~sage.combinat.root_system.type_B.CartanType._latex_dynkin_diagram`.

EXAMPLES::

sage: CartanType(['B',2])._latex_draw_arrow_tip(1, 0, 180)

'\\draw[shift={(1, 0)}, rotate=180] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);\n'

"""

return "\\draw[shift={(%s, %s)}, rotate=%s] (135 : 0.45cm) -- (0,0) -- (-135 : 0.45cm);\n"%(x, y, rot)

@abstract_method

def rank(self):

"""

Return the rank of ``self``.

This is the number of nodes of the associated Coxeter or

Dynkin diagram.

EXAMPLES::

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

sage: CartanType(['A', 7, 2]).rank()

sage: CartanType(['I', 8]).rank()

"""

#return len(self.index_set())

@abstract_method

def index_set(self):

"""

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)

"""

# This coloring scheme is used for crystal graphs and will eventually

# be used for Coxeter groups etc. (experimental feature)

_index_set_coloring = {1:"blue", 2:"red", 3:"green"}

@abstract_method(optional = True)

def coxeter_diagram(self):

"""

Return the Coxeter diagram for ``self``.

EXAMPLES::

sage: CartanType(['B',3]).coxeter_diagram()

Graph on 3 vertices

sage: CartanType(['A',3]).coxeter_diagram().edges()

[(1, 2, 3), (2, 3, 3)]

sage: CartanType(['B',3]).coxeter_diagram().edges()

[(1, 2, 3), (2, 3, 4)]

sage: CartanType(['G',2]).coxeter_diagram().edges()

[(1, 2, 6)]

sage: CartanType(['F',4]).coxeter_diagram().edges()

[(1, 2, 3), (2, 3, 4), (3, 4, 3)]

"""

@cached_method

def coxeter_matrix(self):

"""

Return the Coxeter matrix for ``self``.

EXAMPLES::

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

[1 3 2 2]

[3 1 3 2]

[2 3 1 3]

[2 2 3 1]

"""

from sage.combinat.root_system.coxeter_matrix import CoxeterMatrix

return CoxeterMatrix(self)

def coxeter_type(self):

"""

Return the Coxeter type for ``self``.

EXAMPLES::

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

Coxeter type of ['A', 4]

"""

from sage.combinat.root_system.coxeter_type import CoxeterType

return CoxeterType(self)

def dual(self):

"""

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}

"""

from . import type_dual

return type_dual.CartanType(self)

def relabel(self, 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()

O---O=>=O---O

4 3 2 1

F4 relabelled by {1: 4, 2: 3, 3: 2, 4: 1}

"""

from . import type_relabel

return type_relabel.CartanType(self, relabelling)

def subtype(self, 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()

O=<=O---O=<=O

0 1 2 3

BC3~

sage: ct.subtype([1,2,3])

['C', 3]

"""

return self.cartan_matrix().subtype(index_set).cartan_type()

def marked_nodes(self, 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()

X---O=>=X---O

1 2 3 4

F4 with nodes (1, 3) marked

"""

if not marked_nodes:

return self

from . import type_marked

return type_marked.CartanType(self, marked_nodes)

def is_reducible(self):

"""

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

"""

return not self.is_irreducible()

def is_irreducible(self):

"""

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

"""

return False

def is_atomic(self):

r"""

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 \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

:meth:`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

"""

return self.is_irreducible()

def is_compound(self):

"""

A short hand for not :meth:`is_atomic`.

TESTS::

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

True

"""

return not self.is_atomic()

@abstract_method

def is_finite(self):

"""

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

"""

@abstract_method

def is_affine(self):

"""

Return whether ``self`` is affine.

EXAMPLES::

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

False

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

True

"""

def is_crystallographic(self):

"""

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]]

"""

return False

def is_simply_laced(self):

"""

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]]

"""

return False

def is_implemented(self):

"""

Check whether the Cartan datum for ``self`` is actually implemented.

EXAMPLES::

sage: CartanType(["A",4,1]).is_implemented()

True

sage: CartanType(['H',3]).is_implemented()

True

"""

try:

self.coxeter_diagram()

return True

except Exception:

return False

def root_system(self):

"""

Return the root system associated to ``self``.

EXAMPLES::

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

Root system of type ['A', 4]

"""

from sage.combinat.root_system.root_system import RootSystem

return RootSystem(self)

def as_folding(self, folding_of=None, sigma=None):

r"""

Return ``self`` realized as a folded Cartan type.

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

diagram foldings:

.. MATH::

\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}

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

:class:`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]

"""

from sage.combinat.root_system.type_folded import CartanTypeFolded

if folding_of is None and sigma is None:

return self._default_folded_cartan_type()

if folding_of is None or sigma is None:

raise ValueError("Both folding_of and sigma must be given")

return CartanTypeFolded(self, folding_of, sigma)

def _default_folded_cartan_type(self):

"""

Return the default folded Cartan type.

In general, this just returns ``self`` in ``self`` with `\sigma` as

the identity map.

EXAMPLES::

sage: D = CartanMatrix([[2, -3], [-2, 2]]).dynkin_diagram()

sage: D._default_folded_cartan_type()

Dynkin diagram of rank 2 as a folding of Dynkin diagram of rank 2

"""

from sage.combinat.root_system.type_folded import CartanTypeFolded

return CartanTypeFolded(self, self, [[i] for i in self.index_set()])

options = CartanType.options

class CartanType_crystallographic(CartanType_abstract):

"""

An abstract class for crystallographic Cartan types.

"""

# The default value should really be lambda x:x, but sphinx does

# not like it currently (see #14553); since this is an abstract method

# the value won't actually be used, so we put a fake instead.

@abstract_method(optional=True)

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

r"""

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

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__))

O 1

O---O---O---O=>=O

1 2 2 2 2

"""

# The default value of label should really be lambda i:i, but sphinx does

# not like it currently (see #14553); since this is an abstract method

# the value won't actually be used, so we put a fake instead.

@abstract_method(optional=True)

def _latex_dynkin_diagram(self, label='lambda i: i',

node=None, node_dist=2):

r"""

Return a latex representation of the Dynkin diagram.

INPUT:

- ``label`` -- (default: the identity) a relabeling function

for the nodes

- ``node`` -- (optional) a function which returns the latex for a node

- ``node_dist`` -- (default: 2) the distance between nodes in cm

EXAMPLES::

sage: latex(CartanType(['A',4]).dynkin_diagram()) # indirect doctest

\begin{tikzpicture}[scale=0.5]

\draw (-1,0) node[anchor=east] {$A_{4}$};

\draw (0 cm,0) -- (6 cm,0);

\draw[fill=white] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$1$};

\draw[fill=white] (2 cm, 0 cm) circle (.25cm) node[below=4pt]{$2$};

\draw[fill=white] (4 cm, 0 cm) circle (.25cm) node[below=4pt]{$3$};

\draw[fill=white] (6 cm, 0 cm) circle (.25cm) node[below=4pt]{$4$};

\end{tikzpicture}

"""

@abstract_method

def dynkin_diagram(self):

"""

Return the Dynkin diagram associated with ``self``.

EXAMPLES::

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

O---O---O---O

1 2 3 4

.. NOTE::

Derived subclasses should typically implement this as a cached

method.

"""

@cached_method

def cartan_matrix(self):

"""

Return 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]

"""

from sage.combinat.root_system.cartan_matrix import CartanMatrix

return CartanMatrix(self.dynkin_diagram())

@cached_method

def coxeter_diagram(self):

"""

Return the Coxeter diagram for ``self``.

This implementation constructs it from the Dynkin diagram.

.. SEEALSO:: :meth:`CartanType_abstract.coxeter_diagram`

EXAMPLES::

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

Graph on 3 vertices

sage: CartanType(['A',3]).coxeter_diagram().edges()

[(1, 2, 3), (2, 3, 3)]

sage: CartanType(['B',3]).coxeter_diagram().edges()

[(1, 2, 3), (2, 3, 4)]

sage: CartanType(['G',2]).coxeter_diagram().edges()

[(1, 2, 6)]

sage: CartanType(['F',4]).coxeter_diagram().edges()

[(1, 2, 3), (2, 3, 4), (3, 4, 3)]

sage: CartanType(['A',2,2]).coxeter_diagram().edges()

[(0, 1, +Infinity)]

"""

from sage.rings.infinity import infinity

scalarproducts_to_order = { 0: 2, 1: 3, 2: 4, 3: 6, 4: infinity }

from sage.graphs.graph import Graph

coxeter_diagram = Graph(multiedges=False)

a = self.dynkin_diagram()

I = self.index_set()

coxeter_diagram.add_vertices(I)

for i in I:

for j in a.neighbors_out(i):

# avoid adding the edge twice

if not coxeter_diagram.has_edge(i,j):

coxeter_diagram.add_edge(i,j, scalarproducts_to_order[a[i,j]*a[j,i]])

return coxeter_diagram

def is_crystallographic(self):

"""

Implements :meth:`CartanType_abstract.is_crystallographic`

by returning ``True``.

EXAMPLES::

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

True

"""

return True

@cached_method

def symmetrizer(self):

"""

Return the symmetrizer of the Cartan matrix of ``self``.

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 computes the unique minimal such `D` with positive

integral coefficients. If `D` exists, it is returned as a

family. Otherwise ``None`` is returned.

The coefficients are coerced to ``base_ring``.

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

Here is the symmetrizer of some reducible Cartan types::

sage: T = CartanType(["D", 2])

sage: print(T.ascii_art(T.symmetrizer().__getitem__))

O O

1 1

sage: T = CartanType(["B",5],["BC",5, 2])

sage: print(T.ascii_art(T.symmetrizer().__getitem__))

O---O---O---O=>=O

2 2 2 2 1

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

"""

from sage.matrix.constructor import matrix, diagonal_matrix

m = self.cartan_matrix()

n = m.nrows()

M = matrix(ZZ, n, n*n, sparse = True)

for (i,j) in m.nonzero_positions():

M[i, n * i + j] = m[i,j]

M[j, n * i + j] -= m[j,i]

kern = M.integer_kernel()

c = len(self.dynkin_diagram().connected_components())

if kern.dimension() < c:

# the Cartan matrix is not symmetrizable

return None

assert kern.dimension() == c

# Now the basis contains one vector v per connected component

# C of the Dynkin diagram, or equivalently diagonal block of

# the Cartan matrix. The support of v is exactly that

# connected component, and it symmetrizes the corresponding

# diagonal block of the Cartan matrix. We sum all those vectors.

D = sum(kern.basis())

assert diagonal_matrix(D) * m == m.transpose() * diagonal_matrix(D)

I = self.index_set()

return Family( dict( (I[i], D[i]) for i in range(n) ) )

def index_set_bipartition(self):

r"""

Return a bipartition `\{L,R\}` of the vertices of the Dynkin diagram.

For `i` and `j` both in `L` (or both in `R`), the simple

reflections `s_i` and `s_j` commute.

Of course, the Dynkin diagram should be bipartite. This is

always the case for all finite types.

EXAMPLES::

sage: CartanType(['A',5]).index_set_bipartition()

({1, 3, 5}, {2, 4})

sage: CartanType(['A',2,1]).index_set_bipartition()

Traceback (most recent call last):

...

ValueError: the Dynkin diagram must be bipartite

"""

from sage.graphs.graph import Graph

G = Graph(self.dynkin_diagram())

if not G.is_bipartite():

raise ValueError("the Dynkin diagram must be bipartite")

return G.bipartite_sets()

class CartanType_simply_laced(CartanType_crystallographic):

"""

An abstract class for simply laced Cartan types.

"""

def is_simply_laced(self):

"""

Return whether ``self`` is simply laced, which is ``True``.

EXAMPLES::

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

True

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

True

"""

return True

def dual(self):

"""

Simply laced Cartan types are self-dual, so return ``self``.

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]

"""

return self

class CartanType_simple(CartanType_abstract):

"""

An abstract class for simple Cartan types.

"""

def is_irreducible(self):

"""

Return whether ``self`` is irreducible, which is ``True``.

EXAMPLES::

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

True

"""

return True

class CartanType_finite(CartanType_abstract):

"""

An abstract class for simple affine Cartan types.

"""

def is_finite(self):

"""

EXAMPLES::

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

True

"""

return True

def is_affine(self):

"""

EXAMPLES::

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

False

"""

return False

class CartanType_affine(CartanType_simple, CartanType_crystallographic):

"""

An abstract class for simple affine Cartan types

"""

AmbientSpace = LazyImport('sage.combinat.root_system.type_affine', 'AmbientSpace')

def _ascii_art_node(self, label):

"""

Return the ascii art for the node labeled by ``label``.

EXAMPLES::

sage: ct = CartanType(['A',4,1])

sage: CartanType.options(mark_special_node='both')

sage: ct._ascii_art_node(0)

'@'

sage: CartanType.options._reset()

"""

if (label == self.special_node()

and self.options('mark_special_node') in ['printing', 'both']):

return self.options('special_node_str')

return super(CartanType_affine, self)._ascii_art_node(label)

def _latex_draw_node(self, x, y, label, position="below=4pt"):

r"""

Draw (possibly marked [crossed out]) circular node ``i`` at the

position ``(x,y)`` with node label ``label`` .

- ``position`` -- position of the label relative to the node

- ``anchor`` -- (optional) the anchor point for the label

EXAMPLES::

sage: CartanType.options(mark_special_node='both')

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

'\\draw[fill=black] (0 cm, 0 cm) circle (.25cm) node[below=4pt]{$0$};\n'

sage: CartanType.options._reset()

"""

if label == self.special_node() and self.options('mark_special_node') in ['latex', 'both']:

fill = 'black'

else:

fill = 'white'

return super(CartanType_affine, self)._latex_draw_node(x, y, label, position, fill)

def is_finite(self):

"""

EXAMPLES::

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

False

"""

return False

def is_affine(self):

"""

EXAMPLES::

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

True

"""

return True

def is_untwisted_affine(self):

"""

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

"""

return False

@abstract_method

def special_node(self):

r"""

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 :meth:`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()

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)

"""

@cached_method

def special_nodes(self):

r"""

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

EXAMPLES::

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,)

"""

return tuple(sorted(self.dynkin_diagram().automorphism_group(edge_labels=True).orbit(self.special_node())))

@abstract_method

def classical(self):

r"""

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 :meth:`classical`,

:meth:`sage.combinat.root_system.cartan_type.CartanType_crystallographic.dynkin_diagram`,

and :meth:`.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())

"""

@abstract_method

def basic_untwisted(self):

r"""

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]

"""

def row_annihilator(self, m = None):

r"""

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: 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()

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 :class:`CartanMatrix`.

"""

if m is None:

m = self.cartan_matrix()

if self.index_set() != tuple(range(m.ncols())):

raise NotImplementedError("the Cartan matrix currently must be indexed by [0,1,...,{}]".format(m.ncols()))

annihilator_basis = m.integer_kernel().gens()

if len(annihilator_basis) != 1:

raise ValueError("the kernel is not 1 dimensional")

assert(all(coef > 0 for coef in annihilator_basis[0]))

return Family(dict((i,annihilator_basis[0][i])for i in self.index_set()))

acheck = row_annihilator

def col_annihilator(self):

r"""

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}

``a`` is a shortcut for col_annihilator::

sage: RootSystem(['BC',4,2]).cartan_type().col_annihilator()

Finite family {0: 2, 1: 2, 2: 2, 3: 2, 4: 1}

"""

return self.row_annihilator(self.cartan_matrix().transpose())

a = col_annihilator

def c(self):

r"""

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

:arxiv:`math/0605451`

"""

a = self.a()

acheck = self.acheck()

return Family(dict((i, max(ZZ(1), a[i] // acheck[i]))

for i in self.index_set()))

def translation_factors(self):

r"""

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 :meth:`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()

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])

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

:arxiv:`0804.3781`

"""

a = self.a()

acheck = self.acheck()

if set([1/ZZ(2), 2]).issubset( set(a[i]/acheck[i] for i in self.index_set()) ):

# The test above and the formula below are rather meaningless

# But they detect properly type BC or dual and return the correct value

return Family(dict((i, min(ZZ(1), a[i] / acheck[i]))

for i in self.index_set()))

else:

return self.c()

def _test_dual_classical(self, **options):

r"""

Tests whether the special node of the dual is still the same and whether

the methods dual and classical commute.

TESTS::

sage: C = CartanType(['A',2,2])

sage: C._test_dual_classical()

"""

tester = self._tester(**options)

tester.assertTrue( self.classical().dual() == self.dual().classical() )

tester.assertTrue( self.special_node() == self.dual().special_node() )

def other_affinization(self):

"""

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]

"""

if self.is_untwisted_affine():

result = self.classical().dual().affine().dual()

else:

result = self.dual().classical().dual().affine()

assert result.classical() is self.classical()

return result

##############################################################################

# Concrete base classes

class CartanType_standard(UniqueRepresentation, SageObject):

# Technical methods

def _repr_(self, compact = False):

"""

TESTS::

sage: ct = CartanType(['A',3])

sage: repr(ct)

"['A', 3]"

sage: ct._repr_(compact=True)

'A3'

"""

format = '%s%s' if compact else "['%s', %s]"

return format%(self.letter, self.n)

def __len__(self):

"""

EXAMPLES::

sage: len(CartanType(['A',4]))

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

"""

return 3 if self.is_affine() else 2

def __getitem__(self, i):

"""

EXAMPLES::

sage: t = CartanType(['B', 3])

sage: t[0]

'B'

sage: t[1]

sage: t[2]

Traceback (most recent call last):

...

IndexError: index out of range

"""

if i == 0:

return self.letter

elif i == 1:

return self.n

else:

raise IndexError("index out of range")

class CartanType_standard_finite(CartanType_standard, CartanType_finite):

"""

A concrete base class for the finite standard Cartan types.

This includes for example `A_3`, `D_4`, or `E_8`.

TESTS::

sage: ct1 = CartanType(['A',4])

sage: ct2 = CartanType(['A',4])

sage: ct3 = CartanType(['A',5])

sage: ct1 == ct2

True

sage: ct1 != ct3

True

"""

def __init__(self, letter, n):

"""

EXAMPLES::

sage: ct = CartanType(['A',4])

TESTS::

sage: TestSuite(ct).run(verbose = True)

running ._test_category() . . . pass

running ._test_new() . . . pass

running ._test_not_implemented_methods() . . . pass

running ._test_pickling() . . . pass

"""

# assert(t[0] in ['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I'])

# assert(t[1] in ZZ and t[1] >= 0)

# if t[0] in ['B', 'C']:

# assert(t[1] >= 2)

# if t[0] == 'D':

# assert(t[1] >= 3)

# if t[0] == 'E':

# assert(t[1] <= 8)

# if t[0] == 'F':

# assert(t[1] <= 4)

# if t[0] == 'G':

# assert(t[1] <= 2)

# if t[0] == 'H':

# assert(t[1] <= 4)

self.letter = letter

self.n = n

def __reduce__(self):

"""

TESTS::

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

sage: T.__reduce__()

(CartanType, ('D', 4))

sage: T == loads(dumps(T))

True

"""

from .cartan_type import CartanType

return (CartanType, (self.letter, self.n))

def __hash__(self):

"""

EXAMPLES::

sage: ct = CartanType(['A',2])

sage: hash(ct) #random

-5684143898951441983

"""

return hash((self.n, self.letter))

# mathematical methods

def index_set(self):

"""

Implements :meth:`CartanType_abstract.index_set`.

The index set for all standard finite Cartan types is of the form

`\{1, \ldots, n\}`. (See :mod:`~sage.combinat.root_system.type_I`

for a slight abuse of this).

EXAMPLES::

sage: CartanType(['A', 5]).index_set()

(1, 2, 3, 4, 5)

"""

return tuple(range(1,self.n+1))

def rank(self):

"""

Return the rank of ``self`` which for type `X_n` is `n`.

EXAMPLES::

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

sage: CartanType(['B', 3]).rank()

sage: CartanType(['C', 3]).rank()

sage: CartanType(['D', 4]).rank()

sage: CartanType(['E', 6]).rank()

"""

return self.n

def affine(self):

"""

Return the corresponding untwisted affine Cartan type.

EXAMPLES::

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

['A', 3, 1]

"""

return CartanType([self.letter, self.n, 1])

def coxeter_number(self):

"""

Return the Coxeter number associated with ``self``.

The Coxeter number is the order of a Coxeter element of the

corresponding Weyl group.

See Bourbaki, Lie Groups and Lie Algebras V.6.1 or

:wikipedia:`Coxeter_element` for more information.

EXAMPLES::

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

sage: CartanType(['B',4]).coxeter_number()

sage: CartanType(['C',4]).coxeter_number()

"""

return sum(self.affine().a())

def dual_coxeter_number(self):

"""

Return the Coxeter number associated with ``self``.

EXAMPLES::

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

sage: CartanType(['B',4]).dual_coxeter_number()

sage: CartanType(['C',4]).dual_coxeter_number()

"""

return sum(self.affine().acheck())

def type(self):

"""

Returns the type of ``self``.

EXAMPLES::

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

'A'

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

'A'

"""

return self.letter

@cached_method

def opposition_automorphism(self):

r"""

Returns the opposition automorphism

The *opposition automorphism* is the automorphism

`i \mapsto i^*` of the vertices Dynkin diagram such that,

for `w_0` the longest element of the Weyl group, and any

simple root `\alpha_i`, one has `\alpha_{i^*} = -w_0(\alpha_i)`.

The automorphism is returned as a :class:`Family`.

EXAMPLES::

sage: ct = CartanType(['A', 5])

sage: ct.opposition_automorphism()

Finite family {1: 5, 2: 4, 3: 3, 4: 2, 5: 1}

sage: ct = CartanType(['D', 4])

sage: ct.opposition_automorphism()

Finite family {1: 1, 2: 2, 3: 3, 4: 4}

sage: ct = CartanType(['D', 5])

sage: ct.opposition_automorphism()

Finite family {1: 1, 2: 2, 3: 3, 4: 5, 5: 4}

sage: ct = CartanType(['C', 4])

sage: ct.opposition_automorphism()

Finite family {1: 1, 2: 2, 3: 3, 4: 4}

"""

Q = self.root_system().root_lattice()

W = Q.weyl_group()

w0 = W.long_element()

alpha = Q.simple_roots()

d = {i: (w0.action(alpha[i])).leading_support() for i in self.index_set()}

return Family(d)

##########################################################################

class CartanType_standard_affine(CartanType_standard, CartanType_affine):

r"""

A concrete class for affine simple Cartan types.

"""

def __init__(self, letter, n, affine = 1):

"""

EXAMPLES::

sage: ct = CartanType(['A',4,1])

sage: TestSuite(ct).run()

TESTS::

sage: ct1 = CartanType(['A',3, 1])

sage: ct2 = CartanType(['B',3, 1])

sage: ct3 = CartanType(['A',3])

sage: ct1 == ct1

True

sage: ct1 == ct2

False

sage: ct1 == ct3

False

"""

assert(letter in ['A', 'B', 'C', 'BC', 'D', 'E', 'F', 'G'])

self.letter = letter

self.n = n

self.affine = affine

def _repr_(self, compact = False):

"""

TESTS::

sage: ct = CartanType(['A',3, 1])

sage: repr(ct)

"['A', 3, 1]"

sage: ct._repr_(compact=True)

'A3~'

"""

letter = self.letter

n = self.n

aff = self.affine

if self.options('notation') == "Kac":

if letter == 'BC':

letter = 'A'

n *= 2

if compact:

return '%s%s^%s'%(letter, n, aff)

if compact:

return '%s%s~'%(letter, n)

else:

return "['%s', %s, %s]"%(letter, n, aff)

def __reduce__(self):

"""

TESTS::

sage: T = CartanType(['D', 4, 1])

sage: T.__reduce__()

(CartanType, ('D', 4, 1))

sage: T == loads(dumps(T))

True

"""

from sage.combinat.root_system.cartan_type import CartanType

return (CartanType, (self.letter, self.n, self.affine))

def __getitem__(self, i):

"""

EXAMPLES::

sage: t = CartanType(['A', 3, 1])

sage: t[0]

'A'

sage: t[1]

sage: t[2]

sage: t[3]

Traceback (most recent call last):

...

IndexError: index out of range

"""

if i == 0:

return self.letter

elif i == 1:

return self.n

elif i == 2:

return self.affine

else:

raise IndexError("index out of range")

def rank(self):

"""

Return the rank of ``self`` which for type `X_n^{(1)}` is `n + 1`.

EXAMPLES::

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

sage: CartanType(['B', 4, 1]).rank()

sage: CartanType(['C', 3, 1]).rank()

sage: CartanType(['D', 4, 1]).rank()

sage: CartanType(['E', 6, 1]).rank()

sage: CartanType(['E', 7, 1]).rank()

sage: CartanType(['F', 4, 1]).rank()

sage: CartanType(['G', 2, 1]).rank()

sage: CartanType(['A', 2, 2]).rank()

sage: CartanType(['A', 6, 2]).rank()

sage: CartanType(['A', 7, 2]).rank()

sage: CartanType(['D', 5, 2]).rank()

sage: CartanType(['E', 6, 2]).rank()

sage: CartanType(['D', 4, 3]).rank()

"""

return self.n+1

def index_set(self):

r"""

Implements :meth:`CartanType_abstract.index_set`.

The index set for all standard affine Cartan types is of the form

`\{0, \ldots, n\}`.

EXAMPLES::

sage: CartanType(['A', 5, 1]).index_set()

(0, 1, 2, 3, 4, 5)

"""

return tuple(range(self.n+1))

def special_node(self):

r"""

Implement :meth:`CartanType_abstract.special_node`.

With the standard labelling conventions, `0` is always a

special node.

EXAMPLES::

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

"""

return 0

def type(self):

"""

Return the type of ``self``.

EXAMPLES::

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

'A'

"""

return self.letter

##########################################################################

class CartanType_standard_untwisted_affine(CartanType_standard_affine):

r"""

A concrete class for the standard untwisted affine Cartan types.

"""

def classical(self):

r"""

Return the classical Cartan type associated with ``self``.

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]

"""

return CartanType([self.letter,self.n])

def basic_untwisted(self):

r"""

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]

"""

return self.classical()

def is_untwisted_affine(self):

"""

Implement :meth:`CartanType_affine.is_untwisted_affine` by

returning ``True``.

EXAMPLES::

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

True

"""

return True

def _latex_(self):

r"""

Return a latex representation of ``self``.

EXAMPLES::

sage: latex(CartanType(['B',4,1]))

B_{4}^{(1)}

sage: latex(CartanType(['C',4,1]))

C_{4}^{(1)}

sage: latex(CartanType(['D',4,1]))

D_{4}^{(1)}

sage: latex(CartanType(['F',4,1]))

F_4^{(1)}

sage: latex(CartanType(['G',2,1]))

G_2^{(1)}

"""

return self.classical()._latex_()+"^{(1)}"

##########################################################################

class CartanType_decorator(UniqueRepresentation, SageObject, CartanType_abstract):

"""

Concrete base class for Cartan types that decorate another Cartan type.

"""

def __init__(self, ct):

"""

Initialize ``self``.

EXAMPLES::

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})

sage: TestSuite(ct).run()

"""

self._type = ct

def is_irreducible(self):

"""

EXAMPLES::

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})

sage: ct.is_irreducible()

True

"""

return self._type.is_irreducible()

def is_finite(self):

"""

EXAMPLES::

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})

sage: ct.is_finite()

True

"""

return self._type.is_finite()

def is_crystallographic(self):

"""

EXAMPLES::

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})

sage: ct.is_crystallographic()

True

"""

return self._type.is_crystallographic()

def is_affine(self):

"""

EXAMPLES::

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})

sage: ct.is_affine()

False

"""

return self._type.is_affine()

def rank(self):

"""

EXAMPLES::

sage: ct = CartanType(['G', 2]).relabel({1:2,2:1})

sage: ct.rank()

"""

return self._type.rank()

def index_set(self):

"""

EXAMPLES::

sage: ct = CartanType(['F', 4, 1]).dual()

sage: ct.index_set()

(0, 1, 2, 3, 4)

"""

return self._type.index_set()

##############################################################################

# Base concrete class for superalgebras

class SuperCartanType_standard(UniqueRepresentation, SageObject):

# Technical methods

def _repr_(self, compact = False):

"""

TESTS::

sage: ct = CartanType(['A', [3,2]])

sage: repr(ct)

"['A', [3, 2]]"

sage: ct._repr_(compact=True)

'A3|2'

"""

formatstr = '%s%s|%s' if compact else "['%s', [%s, %s]]"

return formatstr%(self.letter, self.m, self.n)

def __len__(self):

"""

EXAMPLES::

sage: len(CartanType(['A',[4,3]]))

"""

return 2

def __getitem__(self, i):

"""

EXAMPLES::

sage: t = CartanType(['A', [3,6]])

sage: t[0]

'A'

sage: t[1]

[3, 6]

sage: t[2]

Traceback (most recent call last):

...

IndexError: index out of range

"""

if i == 0:

return self.letter

elif i == 1:

return [self.m, self.n]

else:

raise IndexError("index out of range")

options = CartanType.options

##############################################################################

# For backward compatibility

class CartanType_simple_finite(object):

def __setstate__(self, dict):

"""

Implements the unpickling of Cartan types pickled by Sage <= 4.0.

EXAMPLES:

This is the pickle for CartanType(["A", 4])::

sage: pg_CartanType_simple_finite = unpickle_global('sage.combinat.root_system.cartan_type', 'CartanType_simple_finite')

sage: si1 = unpickle_newobj(pg_CartanType_simple_finite, ())

sage: pg_unpickleModule = unpickle_global('twisted.persisted.styles', 'unpickleModule')

sage: pg_make_integer = unpickle_global('sage.rings.integer', 'make_integer')

sage: si2 = pg_make_integer('4')

sage: unpickle_build(si1, {'tools':pg_unpickleModule('sage.combinat.root_system.type_A'), 't':['A', si2], 'letter':'A', 'n':si2})

sage: si1

['A', 4]

sage: si1.dynkin_diagram()

O---O---O---O

1 2 3 4

This is quite hacky; in particular unique representation is not preserved::

sage: si1 == CartanType(["A", 4]) # todo: not implemented

True

"""

T = CartanType([dict['letter'], dict['n']])

self.__class__ = T.__class__

self.__dict__ = T.__dict__

# deprecations from trac:18555

from sage.misc.superseded import deprecated_function_alias

CartanTypeFactory.global_options = deprecated_function_alias(18555, CartanTypeFactory.options)

CartanTypeOptions = deprecated_function_alias(18555, CartanType.options)

CartanType_abstract.global_options = deprecated_function_alias(18555, CartanType.options)

Coverage for local/lib/python2.7/site-packages/sage/combinat/root_system/cartan_type.py : 68%

498 statements 341 run 157 missing 0 excluded