class RootSystem(UniqueRepresentation, SageObject):
r"""
A class for root systems.
EXAMPLES:
We construct the root system for type `B_3`::
sage: R=RootSystem(['B',3]); R
Root system of type ['B', 3]
``R`` models the root system abstractly. It comes equipped with various
realizations of the root and weight lattices, where all computations
take place. Let us play first with the root lattice::
sage: space = R.root_lattice()
sage: space
Root lattice of the Root system of type ['B', 3]
This is the free `\ZZ`-module `\bigoplus_i \ZZ.\alpha_i` spanned
by the simple roots::
sage: space.base_ring()
Integer Ring
sage: list(space.basis())
[alpha[1], alpha[2], alpha[3]]
Let us do some computations with the simple roots::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
alpha[1] + alpha[2]
There is a canonical pairing between the root lattice and the
coroot lattice::
sage: R.coroot_lattice()
Coroot lattice of the Root system of type ['B', 3]
We construct the simple coroots, and do some computations (see
comments about duality below for some caveat)::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
We can carry over the same computations in any of the other
realizations of the root lattice, like the root space
`\bigoplus_i \QQ.\alpha_i`, the weight lattice
`\bigoplus_i \ZZ.\Lambda_i`, the weight
space `\bigoplus_i \QQ.\Lambda_i`. For example::
sage: space = R.weight_space()
sage: space
Weight space over the Rational Field of the Root system of type ['B', 3]
::
sage: space.base_ring()
Rational Field
sage: list(space.basis())
[Lambda[1], Lambda[2], Lambda[3]]
::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
Lambda[1] + Lambda[2] - 2*Lambda[3]
The fundamental weights are the dual basis of the coroots::
sage: Lambda = space.fundamental_weights()
sage: Lambda[1]
Lambda[1]
::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
::
sage: [Lambda[i].scalar(alphacheck[1]) for i in space.index_set()]
[1, 0, 0]
sage: [Lambda[i].scalar(alphacheck[2]) for i in space.index_set()]
[0, 1, 0]
sage: [Lambda[i].scalar(alphacheck[3]) for i in space.index_set()]
[0, 0, 1]
Let us use the simple reflections. In the weight space, they
work as in the *number game*: firing the node `i` on an
element `x` adds `c` times the simple root
`\alpha_i`, where `c` is the coefficient of
`i` in `x`::
sage: s = space.simple_reflections()
sage: Lambda[1].simple_reflection(1)
-Lambda[1] + Lambda[2]
sage: Lambda[2].simple_reflection(1)
Lambda[2]
sage: Lambda[3].simple_reflection(1)
Lambda[3]
sage: (-2*Lambda[1] + Lambda[2] + Lambda[3]).simple_reflection(1)
2*Lambda[1] - Lambda[2] + Lambda[3]
It can be convenient to manipulate the simple reflections
themselves::
sage: s = space.simple_reflections()
sage: s[1](Lambda[1])
-Lambda[1] + Lambda[2]
sage: s[1](Lambda[2])
Lambda[2]
sage: s[1](Lambda[3])
Lambda[3]
.. RUBRIC:: Ambient spaces
The root system may also come equipped with an ambient space.
This is a `\QQ`-module, endowed with its canonical Euclidean
scalar product, which admits simultaneous embeddings of the
(extended) weight and the (extended) coweight lattice, and
therefore the root and the coroot lattice. This is implemented on
a type by type basis for the finite crystalographic root systems
following Bourbaki's conventions and is extended to the affine
cases. Coefficients permitting, this is also available as an
ambient lattice.
.. SEEALSO:: :meth:`ambient_space` and :meth:`ambient_lattice` for details
In finite type `A`, we recover the natural representation of the
symmetric group as group of permutation matrices::
sage: RootSystem(["A",2]).ambient_space().weyl_group().simple_reflections()
Finite family {1: [0 1 0]
[1 0 0]
[0 0 1],
2: [1 0 0]
[0 0 1]
[0 1 0]}
In type `B`, `C`, and `D`, we recover the natural representation
of the Weyl group as groups of signed permutation matrices::
sage: RootSystem(["B",3]).ambient_space().weyl_group().simple_reflections()
Finite family {1: [0 1 0]
[1 0 0]
[0 0 1],
2: [1 0 0]
[0 0 1]
[0 1 0],
3: [ 1 0 0]
[ 0 1 0]
[ 0 0 -1]}
In (untwisted) affine types `A`, ..., `D`, one can recover from
the ambient space the affine permutation representation, in window
notation. Let us consider the ambient space for affine type `A`::
sage: L = RootSystem(["A",2,1]).ambient_space(); L
Ambient space of the Root system of type ['A', 2, 1]
Define the "identity" by an appropriate vector at level -3::
sage: e = L.basis(); Lambda = L.fundamental_weights()
sage: id = e[0] + 2*e[1] + 3*e[2] - 3*Lambda[0]
The corresponding permutation is obtained by projecting it onto
the classical ambient space::
sage: L.classical()
Ambient space of the Root system of type ['A', 2]
sage: L.classical()(id)
(1, 2, 3)
Here is the orbit of the identity under the action of the finite
group::
sage: W = L.weyl_group()
sage: S3 = [ w.action(id) for w in W.classical() ]
sage: [L.classical()(x) for x in S3]
[(1, 2, 3), (2, 1, 3), (3, 1, 2), (3, 2, 1), (1, 3, 2), (2, 3, 1)]
And the action of `s_0` on these yields::
sage: s = W.simple_reflections()
sage: [L.classical()(s[0].action(x)) for x in S3]
[(0, 2, 4), (0, 1, 5), (-1, 1, 6), (-2, 2, 6), (-1, 3, 4), (-2, 3, 5)]
.. RUBRIC:: Dual root systems
The root system is aware of its dual root system::
sage: R.dual
Dual of root system of type ['B', 3]
R.dual is really the root system of type `C_3`::
sage: R.dual.cartan_type()
['C', 3]
And the coroot lattice that we have been manipulating before is
really implemented as the root lattice of the dual root system::
sage: R.dual.root_lattice()
Coroot lattice of the Root system of type ['B', 3]
In particular, the coroots for the root lattice are in fact the
roots of the coroot lattice::
sage: list(R.root_lattice().simple_coroots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.coroot_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.dual.root_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
The coweight lattice and space are defined similarly. Note that, to
limit confusion, all the output have been tweaked appropriately.
.. seealso::
- :mod:`sage.combinat.root_system`
- :class:`RootSpace`
- :class:`WeightSpace`
- :class:`AmbientSpace`
- :class:`~sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations`
- :class:`~sage.combinat.root_system.weight_lattice_realizations.WeightLatticeRealizations`
TESTS::
sage: R = RootSystem(['C',3])
sage: TestSuite(R).run()
sage: L = R.ambient_space()
sage: s = L.simple_reflections() # this used to break the testsuite below due to caching an unpicklable method
sage: s = L.simple_projections() # todo: not implemented
sage: TestSuite(L).run()
sage: L = R.root_space()
sage: s = L.simple_reflections()
sage: TestSuite(L).run()
::
sage: for T in CartanType.samples(crystalographic=True): # long time (13s on sage.math, 2012)
... TestSuite(RootSystem(T)).run()
"""
@staticmethod
def __classcall__(cls, cartan_type, as_dual_of=None):
"""
Straighten arguments to enable unique representation
.. seealso:: :class:`UniqueRepresentation`
TESTS::
sage: RootSystem(["A",3]) is RootSystem(CartanType(["A",3]))
True
sage: RootSystem(["B",3], as_dual_of=None) is RootSystem("B3")
True
"""
return super(RootSystem, cls).__classcall__(cls, CartanType(cartan_type), as_dual_of)
def __init__(self, cartan_type, as_dual_of=None):
"""
TESTS::
sage: R = RootSystem(['A',3])
sage: R
Root system of type ['A', 3]
"""
self._cartan_type = CartanType(cartan_type)
# Duality
# The root system can be defined as dual of another root system. This will
# only affects the pretty printing
if as_dual_of is None:
self.dual_side = False
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
# still fails for CartanType G2xA1
try:
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
except StandardError:
pass
else:
self.dual_side = True
self.dual = as_dual_of
def _test_root_lattice_realizations(self, **options):
"""
Runs tests on all the root lattice realizations of this root
system.
EXAMPLES::
sage: RootSystem(["A",3])._test_root_lattice_realizations()
See also :class:`TestSuite`.
"""
tester = self._tester(**options)
options.pop('tester', None)
from sage.misc.sage_unittest import TestSuite
TestSuite(self.root_lattice()).run(**options)
TestSuite(self.root_space()).run(**options)
TestSuite(self.weight_lattice()).run(**options)
TestSuite(self.weight_space()).run(**options)
if self.cartan_type().is_affine():
TestSuite(self.weight_lattice(extended=True)).run(**options)
TestSuite(self.weight_space(extended=True)).run(**options)
if self.ambient_lattice() is not None:
TestSuite(self.ambient_lattice()).run(**options)
if self.ambient_space() is not None:
TestSuite(self.ambient_space()).run(**options)
def _repr_(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]) # indirect doctest
Root system of type ['A', 3]
sage: RootSystem(['B',3]).dual # indirect doctest
Dual of root system of type ['B', 3]
"""
if self.dual_side:
return "Dual of root system of type %s"%self.dual.cartan_type()
else:
return "Root system of type %s"%self.cartan_type()
def cartan_type(self):
"""
Returns the Cartan type of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.cartan_type()
['A', 3]
"""
return self._cartan_type
@cached_method
def dynkin_diagram(self):
"""
Returns the Dynkin diagram of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.dynkin_diagram()
O---O---O
1 2 3
A3
"""
return self.cartan_type().dynkin_diagram()
@cached_method
def cartan_matrix(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).cartan_matrix()
[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]
"""
return self.cartan_type().cartan_matrix()
def index_set(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).index_set()
[1, 2, 3]
"""
return self.cartan_type().index_set()
@cached_method
def is_finite(self):
"""
Returns True if self is a finite root system.
EXAMPLES::
sage: RootSystem(["A",3]).is_finite()
True
sage: RootSystem(["A",3,1]).is_finite()
False
"""
return self.cartan_type().is_finite()
@cached_method
def is_irreducible(self):
"""
Returns True if self is an irreducible root system.
EXAMPLES::
sage: RootSystem(['A', 3]).is_irreducible()
True
sage: RootSystem("A2xB2").is_irreducible()
False
"""
return self.cartan_type().is_irreducible()
def __cmp__(self, other):
"""
EXAMPLES::
sage: r1 = RootSystem(['A',3])
sage: r2 = RootSystem(['B',3])
sage: r1 == r1
True
sage: r1 == r2
False
"""
if self.__class__ != other.__class__:
return cmp(self.__class__, other.__class__)
if self._cartan_type != other._cartan_type:
return cmp(self._cartan_type, other._cartan_type)
return 0
def root_lattice(self):
"""
Returns the root lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_lattice()
Root lattice of the Root system of type ['A', 3]
"""
return self.root_space(ZZ)
@cached_method
def root_space(self, base_ring=QQ):
"""
Returns the root space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_space()
Root space over the Rational Field of the Root system of type ['A', 3]
"""
return RootSpace(self, base_ring)
def root_poset(self, restricted=False, facade=False):
r"""
Returns the (restricted) root poset associated to ``self``.
The elements are given by the positive roots (resp. non-simple, positive roots), and
`\alpha \leq \beta` iff `\beta - \alpha` is a non-negative linear combination of simple roots.
INPUT:
- ``restricted`` -- (default:False) if True, only non-simple roots are considered.
- ``facade`` -- (default:False) passes facade option to the poset generator.
EXAMPLES::
sage: Phi = RootSystem(['A',2]).root_poset(); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]]]
sage: Phi = RootSystem(['A',3]).root_poset(restricted=True); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1] + alpha[2], alpha[1] + alpha[2] + alpha[3]], [alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
sage: Phi = RootSystem(['B',2]).root_poset(); Phi
Finite poset containing 4 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2], alpha[1] + 2*alpha[2]]]
"""
return self.root_lattice().root_poset(restricted=restricted,facade=facade)
def coroot_lattice(self):
"""
Returns the coroot lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_lattice()
Coroot lattice of the Root system of type ['A', 3]
"""
return self.dual.root_lattice()
def coroot_space(self, base_ring=QQ):
"""
Returns the coroot space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_space()
Coroot space over the Rational Field of the Root system of type ['A', 3]
"""
return self.dual.root_space(base_ring)
@cached_method
def weight_lattice(self, extended = False):
"""
Returns the weight lattice associated to self.
.. see also::
- :meth:`weight_space`
- :meth:`coweight_space`, :meth:`coweight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).weight_lattice()
Weight lattice of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).weight_space(extended = True)
Extended weight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return WeightSpace(self, ZZ, extended = extended)
@cached_method
def weight_space(self, base_ring=QQ, extended = False):
"""
Returns the weight space associated to self.
.. see also::
- :meth:`weight_lattice`
- :meth:`coweight_space`, :meth:`coweight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).weight_space()
Weight space over the Rational Field of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).weight_space(extended = True)
Extended weight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return WeightSpace(self, base_ring, extended = extended)
def coweight_lattice(self, extended = False):
"""
Returns the coweight lattice associated to self.
This is the weight lattice of the dual root system.
.. see also::
- :meth:`coweight_space`
- :meth:`weight_space`, :meth:`weight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).coweight_lattice()
Coweight lattice of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).coweight_lattice(extended = True)
Extended coweight lattice of the Root system of type ['A', 3, 1]
"""
return self.dual.weight_lattice(extended = extended)
def coweight_space(self, base_ring=QQ, extended = False):
"""
Returns the coweight space associated to self.
This is the weight space of the dual root system.
.. see also::
- :meth:`coweight_lattice`
- :meth:`weight_space`, :meth:`weight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).coweight_space()
Coweight space over the Rational Field of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).coweight_space(extended=True)
Extended coweight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return self.dual.weight_space(base_ring, extended = extended)
def ambient_lattice(self):
r"""
Return the ambient lattice for this root_system.
This is the ambient space, over `\ZZ`.
.. SEEALSO::
- :meth:`ambient_space`
- :meth:`root_lattice`
- :meth:`weight_lattice`
EXAMPLES::
sage: RootSystem(['A',4]).ambient_lattice()
Ambient lattice of the Root system of type ['A', 4]
sage: RootSystem(['A',4,1]).ambient_lattice()
Ambient lattice of the Root system of type ['A', 4, 1]
Except in type A, only an ambient space can be realized::
sage: RootSystem(['B',4]).ambient_lattice()
sage: RootSystem(['C',4]).ambient_lattice()
sage: RootSystem(['D',4]).ambient_lattice()
sage: RootSystem(['E',6]).ambient_lattice()
sage: RootSystem(['F',4]).ambient_lattice()
sage: RootSystem(['G',2]).ambient_lattice()
"""
return self.ambient_space(ZZ)
@cached_method
def ambient_space(self, base_ring=QQ):
r"""
Return the usual ambient space for this root_system.
INPUT:
- ``base_ring`` -- a base ring (default: `\QQ`)
This is a ``base_ring``-module, endowed with its canonical
Euclidean scalar product, which admits simultaneous embeddings
into the weight and the coweight lattice, and therefore the
root and the coroot lattice, and preserves scalar products
between elements of the coroot lattice and elements of the
root or weight lattice (and dually).
There is no mechanical way to define the ambient space just
from the Cartan matrix. Instead is is constructed from hard
coded type by type data, according to the usual Bourbaki
conventions. Such data is provided for all the finite
(crystalographic) types. From this data, ambient spaces can be
built as well for dual types, reducible types and affine
types. When no data is available, or if the base ring is not
large enough, None is returned.
.. WARNING:: for affine types
.. SEEALSO::
- The section on ambient spaces in :class:`RootSystem`
- :meth:`ambient_lattice`
- :class:`~sage.combinat.root_system.ambient_space.AmbientSpace`
- :class:`~sage.combinat.root_system.ambient_space.type_affine.AmbientSpace`
- :meth:`root_space`
- :meth:`weight:space`
EXAMPLES::
sage: RootSystem(['A',4]).ambient_space()
Ambient space of the Root system of type ['A', 4]
::
sage: RootSystem(['B',4]).ambient_space()
Ambient space of the Root system of type ['B', 4]
::
sage: RootSystem(['C',4]).ambient_space()
Ambient space of the Root system of type ['C', 4]
::
sage: RootSystem(['D',4]).ambient_space()
Ambient space of the Root system of type ['D', 4]
::
sage: RootSystem(['E',6]).ambient_space()
Ambient space of the Root system of type ['E', 6]
::
sage: RootSystem(['F',4]).ambient_space()
Ambient space of the Root system of type ['F', 4]
::
sage: RootSystem(['G',2]).ambient_space()
Ambient space of the Root system of type ['G', 2]
An alternative base ring can be provided as an option::
sage: e = RootSystem(['B',3]).ambient_space(RR)
sage: TestSuite(e).run()
It should contain the smallest ring over which the ambient
space can be defined (`\ZZ` in type `A` or `\QQ` otherwise).
Otherwise ``None`` is returned::
sage: RootSystem(['B',2]).ambient_space(ZZ)
The base ring should also be totally ordered. In practice,
only `\ZZ` and `\QQ` are really supported at this point, but
you are welcome to experiment::
sage: e = RootSystem(['G',2]).ambient_space(RR)
sage: TestSuite(e).run()
Failure in _test_root_lattice_realization:
Traceback (most recent call last):
...
AssertionError: 2.00000000000000 != 2.00000000000000
------------------------------------------------------------
The following tests failed: _test_root_lattice_realization
"""
if not hasattr(self.cartan_type(),"AmbientSpace"):
return None
AmbientSpace = self.cartan_type().AmbientSpace
if not base_ring.has_coerce_map_from(AmbientSpace.smallest_base_ring(self.cartan_type())):
return None
return AmbientSpace(self, base_ring)
class RootSystem(UniqueRepresentation, SageObject):
r"""
A class for root systems.
EXAMPLES:
We construct the root system for type `B_3`::
sage: R=RootSystem(['B',3]); R
Root system of type ['B', 3]
``R`` models the root system abstractly. It comes equipped with various
realizations of the root and weight lattices, where all computation
take place. Let us play first with the root lattice::
sage: space = R.root_lattice()
sage: space
Root lattice of the Root system of type ['B', 3]
It is the free `\ZZ`-module
`\bigoplus_i \ZZ.\alpha_i` spanned by the simple
roots::
sage: space.base_ring()
Integer Ring
sage: list(space.basis())
[alpha[1], alpha[2], alpha[3]]
Let us do some computations with the simple roots::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
alpha[1] + alpha[2]
There is a canonical pairing between the root lattice and the
coroot lattice::
sage: R.coroot_lattice()
Coroot lattice of the Root system of type ['B', 3]
We construct the simple coroots, and do some computations (see
comments about duality below for some caveat)::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
We can carry over the same computations in any of the other
realizations of the root lattice, like the root space
`\bigoplus_i \QQ.\alpha_i`, the weight lattice
`\bigoplus_i \ZZ.\Lambda_i`, the weight
space `\bigoplus_i \QQ.\Lambda_i`. For example::
sage: space = R.weight_space()
sage: space
Weight space over the Rational Field of the Root system of type ['B', 3]
::
sage: space.base_ring()
Rational Field
sage: list(space.basis())
[Lambda[1], Lambda[2], Lambda[3]]
::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
Lambda[1] + Lambda[2] - 2*Lambda[3]
The fundamental weights are the dual basis of the coroots::
sage: Lambda = space.fundamental_weights()
sage: Lambda[1]
Lambda[1]
::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
::
sage: [Lambda[i].scalar(alphacheck[1]) for i in space.index_set()]
[1, 0, 0]
sage: [Lambda[i].scalar(alphacheck[2]) for i in space.index_set()]
[0, 1, 0]
sage: [Lambda[i].scalar(alphacheck[3]) for i in space.index_set()]
[0, 0, 1]
Let us use the simple reflections. In the weight space, they
work as in the *number game*: firing the node `i` on an
element `x` adds `c` times the simple root
`\alpha_i`, where `c` is the coefficient of
`i` in `x`::
sage: s = space.simple_reflections()
sage: Lambda[1].simple_reflection(1)
-Lambda[1] + Lambda[2]
sage: Lambda[2].simple_reflection(1)
Lambda[2]
sage: Lambda[3].simple_reflection(1)
Lambda[3]
sage: (-2*Lambda[1] + Lambda[2] + Lambda[3]).simple_reflection(1)
2*Lambda[1] - Lambda[2] + Lambda[3]
It can be convenient to manipulate the simple reflections
themselves::
sage: s = space.simple_reflections()
sage: s[1](Lambda[1])
-Lambda[1] + Lambda[2]
sage: s[1](Lambda[2])
Lambda[2]
sage: s[1](Lambda[3])
Lambda[3]
The root system may also come equipped with an ambient space, that
is a simultaneous realization of the weight lattice and the coroot
lattice in a Euclidean vector space. This is implemented on a type
by type basis, and is not always available. When the coefficients
permit it, this is also available as an ambient lattice.
TODO: Demo: signed permutations realization of type B
The root system is aware of its dual root system::
sage: R.dual
Dual of root system of type ['B', 3]
R.dual is really the root system of type `C_3`::
sage: R.dual.cartan_type()
['C', 3]
And the coroot lattice that we have been manipulating before is
really implemented as the root lattice of the dual root system::
sage: R.dual.root_lattice()
Coroot lattice of the Root system of type ['B', 3]
In particular, the coroots for the root lattice are in fact the
roots of the coroot lattice::
sage: list(R.root_lattice().simple_coroots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.coroot_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.dual.root_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
The coweight lattice and space are defined similarly. Note that, to
limit confusion, all the output have been tweaked appropriately.
TESTS::
sage: R = RootSystem(['C',3])
sage: R == loads(dumps(R))
True
sage: L = R.ambient_space()
sage: s = L.simple_reflections()
sage: s = L.simple_projections() # todo: not implemented
sage: L == loads(dumps(L))
True
sage: L = R.root_space()
sage: s = L.simple_reflections()
sage: L == loads(dumps(L))
True
::
sage: for T in CartanType.samples(finite=True,crystalographic=True):
... TestSuite(RootSystem(T)).run()
"""
@staticmethod
def __classcall__(cls, cartan_type, as_dual_of=None):
return super(RootSystem, cls).__classcall__(cls, CartanType(cartan_type), as_dual_of)
def __init__(self, cartan_type, as_dual_of=None):
"""
TESTS::
sage: R = RootSystem(['A',3])
sage: R
Root system of type ['A', 3]
"""
self._cartan_type = CartanType(cartan_type)
# Duality
# The root system can be defined as dual of another root system. This will
# only affects the pretty printing
if as_dual_of is None:
self.dual_side = False
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
# still fails for CartanType G2xA1
try:
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
except:
pass
else:
self.dual_side = True
self.dual = as_dual_of
def _test_root_lattice_realizations(self, **options):
"""
Runs tests on all the root lattice realizations of this root
system.
EXAMPLES::
sage: RootSystem(["A",3])._test_root_lattice_realizations()
See also :class:`TestSuite`.
"""
tester = self._tester(**options)
options.pop('tester', None)
from sage.misc.sage_unittest import TestSuite
TestSuite(self.root_lattice()).run(**options)
TestSuite(self.root_space()).run(**options)
TestSuite(self.weight_lattice()).run(**options)
TestSuite(self.weight_space()).run(**options)
if self.ambient_lattice() is not None:
TestSuite(self.ambient_lattice()).run(**options)
if self.ambient_space() is not None:
TestSuite(self.ambient_space()).run(**options)
def _repr_(self):
"""
EXAMPLES::
sage: RootSystem(['A',3])
Root system of type ['A', 3]
"""
if self.dual_side:
return "Dual of root system of type %s"%self.dual.cartan_type()
else:
return "Root system of type %s"%self.cartan_type()
def cartan_type(self):
"""
Returns the Cartan type of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.cartan_type()
['A', 3]
"""
return self._cartan_type
@cached_method
def dynkin_diagram(self):
"""
Returns the Dynkin diagram of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.dynkin_diagram()
O---O---O
1 2 3
A3
"""
return self.cartan_type().dynkin_diagram()
@cached_method
def cartan_matrix(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).cartan_matrix()
[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]
"""
return self.cartan_type().cartan_matrix()
def index_set(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).index_set()
[1, 2, 3]
"""
return self.cartan_type().index_set()
@cached_method
def is_finite(self):
"""
Returns True if self is a finite root system.
EXAMPLES::
sage: RootSystem(["A",3]).is_finite()
True
sage: RootSystem(["A",3,1]).is_finite()
False
"""
return self.cartan_type().is_finite()
@cached_method
def is_irreducible(self):
"""
Returns True if self is an irreducible root system.
EXAMPLES::
sage: RootSystem(['A', 3]).is_irreducible()
True
sage: RootSystem("A2xB2").is_irreducible()
False
"""
return self.cartan_type().is_irreducible()
def __cmp__(self, other):
"""
EXAMPLES::
sage: r1 = RootSystem(['A',3])
sage: r2 = RootSystem(['B',3])
sage: r1 == r1
True
sage: r1 == r2
False
"""
if self.__class__ != other.__class__:
return cmp(self.__class__, other.__class__)
if self._cartan_type != other._cartan_type:
return cmp(self._cartan_type, other._cartan_type)
return 0
def root_lattice(self):
"""
Returns the root lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_lattice()
Root lattice of the Root system of type ['A', 3]
"""
return self.root_space(ZZ)
@cached_method
def root_space(self, base_ring=QQ):
"""
Returns the root space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_space()
Root space over the Rational Field of the Root system of type ['A', 3]
"""
return RootSpace(self, base_ring)
def root_poset(self, restricted=False):
r"""
Returns the (restricted) root poset associated to ``self``.
The elements are given by the positive roots (resp. non-simple, positive roots), and
`\alpha \leq \beta` iff `\beta - \alpha` is a non-negative linear combination of simple roots.
INPUT:
- restricted -- (default:False) if True, only non-simple roots are considered.
EXAMPLES::
sage: Phi = RootSystem(['A',2]).root_poset(); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]]]
sage: Phi = RootSystem(['A',3]).root_poset(restricted=True); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1] + alpha[2], alpha[1] + alpha[2] + alpha[3]], [alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
sage: Phi = RootSystem(['B',2]).root_poset(); Phi
Finite poset containing 4 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2], alpha[1] + 2*alpha[2]]]
"""
return self.root_lattice().root_poset(restricted=restricted)
def coroot_lattice(self):
"""
Returns the coroot lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_lattice()
Coroot lattice of the Root system of type ['A', 3]
"""
return self.dual.root_lattice()
def coroot_space(self, base_ring=QQ):
"""
Returns the coroot space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_space()
Coroot space over the Rational Field of the Root system of type ['A', 3]
"""
return self.dual.root_space(base_ring)
def weight_lattice(self):
"""
Returns the weight lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).weight_lattice()
Weight lattice of the Root system of type ['A', 3]
"""
return self.weight_space(ZZ)
@cached_method
def weight_space(self, base_ring=QQ):
"""
Returns the weight space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).weight_space()
Weight space over the Rational Field of the Root system of type ['A', 3]
"""
return WeightSpace(self, base_ring)
def coweight_lattice(self):
"""
Returns the coweight lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coweight_lattice()
Coweight lattice of the Root system of type ['A', 3]
"""
return self.dual.weight_lattice()
def coweight_space(self, base_ring=QQ):
"""
Returns the weight space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coweight_space()
Coweight space over the Rational Field of the Root system of type ['A', 3]
"""
return self.dual.weight_space(base_ring)
def ambient_lattice(self):
r"""
Returns the usual ambient lattice for this root_system, if it
exists and is implemented, and None otherwise. This is a
`\ZZ`-module, endowed with its canonical euclidean
scalar product, which embeds simultaneously the root lattice
and the coroot lattice (what about the weight lattice?)
EXAMPLES::
sage: RootSystem(['A',4]).ambient_lattice()
Ambient lattice of the Root system of type ['A', 4]
::
sage: RootSystem(['B',4]).ambient_lattice()
sage: RootSystem(['C',4]).ambient_lattice()
sage: RootSystem(['D',4]).ambient_lattice()
sage: RootSystem(['E',6]).ambient_lattice()
sage: RootSystem(['F',4]).ambient_lattice()
sage: RootSystem(['G',2]).ambient_lattice()
"""
return self.ambient_space(ZZ)
@cached_method
def ambient_space(self, base_ring=QQ):
r"""
Returns the usual ambient space for this root_system, if it is
implemented, and None otherwise. This is a `\QQ`-module, endowed with
its canonical euclidean scalar product, which embeds simultaneously
the root lattice and the coroot lattice (what about the weight
lattice?). An alternative base ring can be provided as an option;
it must contain the smallest ring over which the ambient space can
be defined (`\ZZ` or `\QQ`, depending on the type).
EXAMPLES::
sage: RootSystem(['A',4]).ambient_space()
Ambient space of the Root system of type ['A', 4]
::
sage: RootSystem(['B',4]).ambient_space()
Ambient space of the Root system of type ['B', 4]
::
sage: RootSystem(['C',4]).ambient_space()
Ambient space of the Root system of type ['C', 4]
::
sage: RootSystem(['D',4]).ambient_space()
Ambient space of the Root system of type ['D', 4]
::
sage: RootSystem(['E',6]).ambient_space()
Ambient space of the Root system of type ['E', 6]
::
sage: RootSystem(['F',4]).ambient_space()
Ambient space of the Root system of type ['F', 4]
::
sage: RootSystem(['G',2]).ambient_space()
Ambient space of the Root system of type ['G', 2]
"""
# Intention: check that the ambient_space is implemented and that
# base_ring contains the smallest base ring for this ambient space
if not hasattr(self.cartan_type(),"AmbientSpace"):
return None
AmbientSpace = self.cartan_type().AmbientSpace
if base_ring == ZZ and AmbientSpace.smallest_base_ring() == QQ:
return None
return AmbientSpace(self, base_ring)
class RootSystem(UniqueRepresentation, SageObject):
r"""
A class for root systems.
EXAMPLES:
We construct the root system for type `B_3`::
sage: R=RootSystem(['B',3]); R
Root system of type ['B', 3]
``R`` models the root system abstractly. It comes equipped with various
realizations of the root and weight lattices, where all computations
take place. Let us play first with the root lattice::
sage: space = R.root_lattice()
sage: space
Root lattice of the Root system of type ['B', 3]
This is the free `\ZZ`-module `\bigoplus_i \ZZ.\alpha_i` spanned
by the simple roots::
sage: space.base_ring()
Integer Ring
sage: list(space.basis())
[alpha[1], alpha[2], alpha[3]]
Let us do some computations with the simple roots::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
alpha[1] + alpha[2]
There is a canonical pairing between the root lattice and the
coroot lattice::
sage: R.coroot_lattice()
Coroot lattice of the Root system of type ['B', 3]
We construct the simple coroots, and do some computations (see
comments about duality below for some caveat)::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
We can carry over the same computations in any of the other
realizations of the root lattice, like the root space
`\bigoplus_i \QQ.\alpha_i`, the weight lattice
`\bigoplus_i \ZZ.\Lambda_i`, the weight
space `\bigoplus_i \QQ.\Lambda_i`. For example::
sage: space = R.weight_space()
sage: space
Weight space over the Rational Field of the Root system of type ['B', 3]
::
sage: space.base_ring()
Rational Field
sage: list(space.basis())
[Lambda[1], Lambda[2], Lambda[3]]
::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
Lambda[1] + Lambda[2] - 2*Lambda[3]
The fundamental weights are the dual basis of the coroots::
sage: Lambda = space.fundamental_weights()
sage: Lambda[1]
Lambda[1]
::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
::
sage: [Lambda[i].scalar(alphacheck[1]) for i in space.index_set()]
[1, 0, 0]
sage: [Lambda[i].scalar(alphacheck[2]) for i in space.index_set()]
[0, 1, 0]
sage: [Lambda[i].scalar(alphacheck[3]) for i in space.index_set()]
[0, 0, 1]
Let us use the simple reflections. In the weight space, they
work as in the *number game*: firing the node `i` on an
element `x` adds `c` times the simple root
`\alpha_i`, where `c` is the coefficient of
`i` in `x`::
sage: s = space.simple_reflections()
sage: Lambda[1].simple_reflection(1)
-Lambda[1] + Lambda[2]
sage: Lambda[2].simple_reflection(1)
Lambda[2]
sage: Lambda[3].simple_reflection(1)
Lambda[3]
sage: (-2*Lambda[1] + Lambda[2] + Lambda[3]).simple_reflection(1)
2*Lambda[1] - Lambda[2] + Lambda[3]
It can be convenient to manipulate the simple reflections
themselves::
sage: s = space.simple_reflections()
sage: s[1](Lambda[1])
-Lambda[1] + Lambda[2]
sage: s[1](Lambda[2])
Lambda[2]
sage: s[1](Lambda[3])
Lambda[3]
.. RUBRIC:: Ambient spaces
The root system may also come equipped with an ambient space.
This is a `\QQ`-module, endowed with its canonical Euclidean
scalar product, which admits simultaneous embeddings of the
(extended) weight and the (extended) coweight lattice, and
therefore the root and the coroot lattice. This is implemented on
a type by type basis for the finite crystalographic root systems
following Bourbaki's conventions and is extended to the affine
cases. Coefficients permitting, this is also available as an
ambient lattice.
.. SEEALSO:: :meth:`ambient_space` and :meth:`ambient_lattice` for details
In finite type `A`, we recover the natural representation of the
symmetric group as group of permutation matrices::
sage: RootSystem(["A",2]).ambient_space().weyl_group().simple_reflections()
Finite family {1: [0 1 0]
[1 0 0]
[0 0 1],
2: [1 0 0]
[0 0 1]
[0 1 0]}
In type `B`, `C`, and `D`, we recover the natural representation
of the Weyl group as groups of signed permutation matrices::
sage: RootSystem(["B",3]).ambient_space().weyl_group().simple_reflections()
Finite family {1: [0 1 0]
[1 0 0]
[0 0 1],
2: [1 0 0]
[0 0 1]
[0 1 0],
3: [ 1 0 0]
[ 0 1 0]
[ 0 0 -1]}
In (untwisted) affine types `A`, ..., `D`, one can recover from
the ambient space the affine permutation representation, in window
notation. Let us consider the ambient space for affine type `A`::
sage: L = RootSystem(["A",2,1]).ambient_space(); L
Ambient space of the Root system of type ['A', 2, 1]
Define the "identity" by an appropriate vector at level -3::
sage: e = L.basis(); Lambda = L.fundamental_weights()
sage: id = e[0] + 2*e[1] + 3*e[2] - 3*Lambda[0]
The corresponding permutation is obtained by projecting it onto
the classical ambient space::
sage: L.classical()
Ambient space of the Root system of type ['A', 2]
sage: L.classical()(id)
(1, 2, 3)
Here is the orbit of the identity under the action of the finite
group::
sage: W = L.weyl_group()
sage: S3 = [ w.action(id) for w in W.classical() ]
sage: [L.classical()(x) for x in S3]
[(1, 2, 3), (2, 1, 3), (3, 1, 2), (3, 2, 1), (1, 3, 2), (2, 3, 1)]
And the action of `s_0` on these yields::
sage: s = W.simple_reflections()
sage: [L.classical()(s[0].action(x)) for x in S3]
[(0, 2, 4), (0, 1, 5), (-1, 1, 6), (-2, 2, 6), (-1, 3, 4), (-2, 3, 5)]
.. RUBRIC:: Dual root systems
The root system is aware of its dual root system::
sage: R.dual
Dual of root system of type ['B', 3]
R.dual is really the root system of type `C_3`::
sage: R.dual.cartan_type()
['C', 3]
And the coroot lattice that we have been manipulating before is
really implemented as the root lattice of the dual root system::
sage: R.dual.root_lattice()
Coroot lattice of the Root system of type ['B', 3]
In particular, the coroots for the root lattice are in fact the
roots of the coroot lattice::
sage: list(R.root_lattice().simple_coroots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.coroot_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.dual.root_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
The coweight lattice and space are defined similarly. Note that, to
limit confusion, all the output have been tweaked appropriately.
.. seealso::
- :mod:`sage.combinat.root_system`
- :class:`RootSpace`
- :class:`WeightSpace`
- :class:`AmbientSpace`
- :class:`~sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations`
- :class:`~sage.combinat.root_system.weight_lattice_realizations.WeightLatticeRealizations`
TESTS::
sage: R = RootSystem(['C',3])
sage: TestSuite(R).run()
sage: L = R.ambient_space()
sage: s = L.simple_reflections() # this used to break the testsuite below due to caching an unpicklable method
sage: s = L.simple_projections() # todo: not implemented
sage: TestSuite(L).run()
sage: L = R.root_space()
sage: s = L.simple_reflections()
sage: TestSuite(L).run()
::
sage: for T in CartanType.samples(crystalographic=True): # long time (13s on sage.math, 2012)
... TestSuite(RootSystem(T)).run()
"""
@staticmethod
def __classcall__(cls, cartan_type, as_dual_of=None):
"""
Straighten arguments to enable unique representation
.. seealso:: :class:`UniqueRepresentation`
TESTS::
sage: RootSystem(["A",3]) is RootSystem(CartanType(["A",3]))
True
sage: RootSystem(["B",3], as_dual_of=None) is RootSystem("B3")
True
"""
return super(RootSystem, cls).__classcall__(cls, CartanType(cartan_type), as_dual_of)
def __init__(self, cartan_type, as_dual_of=None):
"""
TESTS::
sage: R = RootSystem(['A',3])
sage: R
Root system of type ['A', 3]
"""
self._cartan_type = CartanType(cartan_type)
# Duality
# The root system can be defined as dual of another root system. This will
# only affects the pretty printing
if as_dual_of is None:
self.dual_side = False
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
# still fails for CartanType G2xA1
try:
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
except StandardError:
pass
else:
self.dual_side = True
self.dual = as_dual_of
def _test_root_lattice_realizations(self, **options):
"""
Runs tests on all the root lattice realizations of this root
system.
EXAMPLES::
sage: RootSystem(["A",3])._test_root_lattice_realizations()
See also :class:`TestSuite`.
"""
tester = self._tester(**options)
options.pop('tester', None)
from sage.misc.sage_unittest import TestSuite
TestSuite(self.root_lattice()).run(**options)
TestSuite(self.root_space()).run(**options)
TestSuite(self.weight_lattice()).run(**options)
TestSuite(self.weight_space()).run(**options)
if self.cartan_type().is_affine():
TestSuite(self.weight_lattice(extended=True)).run(**options)
TestSuite(self.weight_space(extended=True)).run(**options)
if self.ambient_lattice() is not None:
TestSuite(self.ambient_lattice()).run(**options)
if self.ambient_space() is not None:
TestSuite(self.ambient_space()).run(**options)
def _repr_(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]) # indirect doctest
Root system of type ['A', 3]
sage: RootSystem(['B',3]).dual # indirect doctest
Dual of root system of type ['B', 3]
"""
if self.dual_side:
return "Dual of root system of type %s"%self.dual.cartan_type()
else:
return "Root system of type %s"%self.cartan_type()
def cartan_type(self):
"""
Returns the Cartan type of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.cartan_type()
['A', 3]
"""
return self._cartan_type
@cached_method
def dynkin_diagram(self):
"""
Returns the Dynkin diagram of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.dynkin_diagram()
O---O---O
1 2 3
A3
"""
return self.cartan_type().dynkin_diagram()
@cached_method
def cartan_matrix(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).cartan_matrix()
[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]
"""
return self.cartan_type().cartan_matrix()
def index_set(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).index_set()
[1, 2, 3]
"""
return self.cartan_type().index_set()
@cached_method
def is_finite(self):
"""
Returns True if self is a finite root system.
EXAMPLES::
sage: RootSystem(["A",3]).is_finite()
True
sage: RootSystem(["A",3,1]).is_finite()
False
"""
return self.cartan_type().is_finite()
@cached_method
def is_irreducible(self):
"""
Returns True if self is an irreducible root system.
EXAMPLES::
sage: RootSystem(['A', 3]).is_irreducible()
True
sage: RootSystem("A2xB2").is_irreducible()
False
"""
return self.cartan_type().is_irreducible()
def __cmp__(self, other):
"""
EXAMPLES::
sage: r1 = RootSystem(['A',3])
sage: r2 = RootSystem(['B',3])
sage: r1 == r1
True
sage: r1 == r2
False
"""
if self.__class__ != other.__class__:
return cmp(self.__class__, other.__class__)
if self._cartan_type != other._cartan_type:
return cmp(self._cartan_type, other._cartan_type)
return 0
def root_lattice(self):
"""
Returns the root lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_lattice()
Root lattice of the Root system of type ['A', 3]
"""
return self.root_space(ZZ)
@cached_method
def root_space(self, base_ring=QQ):
"""
Returns the root space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_space()
Root space over the Rational Field of the Root system of type ['A', 3]
"""
return RootSpace(self, base_ring)
def root_poset(self, restricted=False, facade=False):
r"""
Returns the (restricted) root poset associated to ``self``.
The elements are given by the positive roots (resp. non-simple, positive roots), and
`\alpha \leq \beta` iff `\beta - \alpha` is a non-negative linear combination of simple roots.
INPUT:
- ``restricted`` -- (default:False) if True, only non-simple roots are considered.
- ``facade`` -- (default:False) passes facade option to the poset generator.
EXAMPLES::
sage: Phi = RootSystem(['A',2]).root_poset(); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]]]
sage: Phi = RootSystem(['A',3]).root_poset(restricted=True); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1] + alpha[2], alpha[1] + alpha[2] + alpha[3]], [alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
sage: Phi = RootSystem(['B',2]).root_poset(); Phi
Finite poset containing 4 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2], alpha[1] + 2*alpha[2]]]
"""
return self.root_lattice().root_poset(restricted=restricted,facade=facade)
def coroot_lattice(self):
"""
Returns the coroot lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_lattice()
Coroot lattice of the Root system of type ['A', 3]
"""
return self.dual.root_lattice()
def coroot_space(self, base_ring=QQ):
"""
Returns the coroot space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_space()
Coroot space over the Rational Field of the Root system of type ['A', 3]
"""
return self.dual.root_space(base_ring)
@cached_method
def weight_lattice(self, extended = False):
"""
Returns the weight lattice associated to self.
.. see also::
- :meth:`weight_space`
- :meth:`coweight_space`, :meth:`coweight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).weight_lattice()
Weight lattice of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).weight_space(extended = True)
Extended weight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return WeightSpace(self, ZZ, extended = extended)
@cached_method
def weight_space(self, base_ring=QQ, extended = False):
"""
Returns the weight space associated to self.
.. see also::
- :meth:`weight_lattice`
- :meth:`coweight_space`, :meth:`coweight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).weight_space()
Weight space over the Rational Field of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).weight_space(extended = True)
Extended weight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return WeightSpace(self, base_ring, extended = extended)
def coweight_lattice(self, extended = False):
"""
Returns the coweight lattice associated to self.
This is the weight lattice of the dual root system.
.. see also::
- :meth:`coweight_space`
- :meth:`weight_space`, :meth:`weight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).coweight_lattice()
Coweight lattice of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).coweight_lattice(extended = True)
Extended coweight lattice of the Root system of type ['A', 3, 1]
"""
return self.dual.weight_lattice(extended = extended)
def coweight_space(self, base_ring=QQ, extended = False):
"""
Returns the coweight space associated to self.
This is the weight space of the dual root system.
.. see also::
- :meth:`coweight_lattice`
- :meth:`weight_space`, :meth:`weight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).coweight_space()
Coweight space over the Rational Field of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).coweight_space(extended=True)
Extended coweight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return self.dual.weight_space(base_ring, extended = extended)
def ambient_lattice(self):
r"""
Return the ambient lattice for this root_system.
This is the ambient space, over `\ZZ`.
.. SEEALSO::
- :meth:`ambient_space`
- :meth:`root_lattice`
- :meth:`weight_lattice`
EXAMPLES::
sage: RootSystem(['A',4]).ambient_lattice()
Ambient lattice of the Root system of type ['A', 4]
sage: RootSystem(['A',4,1]).ambient_lattice()
Ambient lattice of the Root system of type ['A', 4, 1]
Except in type A, only an ambient space can be realized::
sage: RootSystem(['B',4]).ambient_lattice()
sage: RootSystem(['C',4]).ambient_lattice()
sage: RootSystem(['D',4]).ambient_lattice()
sage: RootSystem(['E',6]).ambient_lattice()
sage: RootSystem(['F',4]).ambient_lattice()
sage: RootSystem(['G',2]).ambient_lattice()
"""
return self.ambient_space(ZZ)
@cached_method
def ambient_space(self, base_ring=QQ):
r"""
Return the usual ambient space for this root_system.
INPUT:
- ``base_ring`` -- a base ring (default: `\QQ`)
This is a ``base_ring``-module, endowed with its canonical
Euclidean scalar product, which admits simultaneous embeddings
into the weight and the coweight lattice, and therefore the
root and the coroot lattice, and preserves scalar products
between elements of the coroot lattice and elements of the
root or weight lattice (and dually).
There is no mechanical way to define the ambient space just
from the Cartan matrix. Instead is is constructed from hard
coded type by type data, according to the usual Bourbaki
conventions. Such data is provided for all the finite
(crystalographic) types. From this data, ambient spaces can be
built as well for dual types, reducible types and affine
types. When no data is available, or if the base ring is not
large enough, None is returned.
.. WARNING:: for affine types
.. SEEALSO::
- The section on ambient spaces in :class:`RootSystem`
- :meth:`ambient_lattice`
- :class:`~sage.combinat.root_system.ambient_space.AmbientSpace`
- :class:`~sage.combinat.root_system.ambient_space.type_affine.AmbientSpace`
- :meth:`root_space`
- :meth:`weight:space`
EXAMPLES::
sage: RootSystem(['A',4]).ambient_space()
Ambient space of the Root system of type ['A', 4]
::
sage: RootSystem(['B',4]).ambient_space()
Ambient space of the Root system of type ['B', 4]
::
sage: RootSystem(['C',4]).ambient_space()
Ambient space of the Root system of type ['C', 4]
::
sage: RootSystem(['D',4]).ambient_space()
Ambient space of the Root system of type ['D', 4]
::
sage: RootSystem(['E',6]).ambient_space()
Ambient space of the Root system of type ['E', 6]
::
sage: RootSystem(['F',4]).ambient_space()
Ambient space of the Root system of type ['F', 4]
::
sage: RootSystem(['G',2]).ambient_space()
Ambient space of the Root system of type ['G', 2]
An alternative base ring can be provided as an option::
sage: e = RootSystem(['B',3]).ambient_space(RR)
sage: TestSuite(e).run()
It should contain the smallest ring over which the ambient
space can be defined (`\ZZ` in type `A` or `\QQ` otherwise).
Otherwise ``None`` is returned::
sage: RootSystem(['B',2]).ambient_space(ZZ)
The base ring should also be totally ordered. In practice,
only `\ZZ` and `\QQ` are really supported at this point, but
you are welcome to experiment::
sage: e = RootSystem(['G',2]).ambient_space(RR)
sage: TestSuite(e).run()
Failure in _test_root_lattice_realization:
Traceback (most recent call last):
...
AssertionError: 2.00000000000000 != 2.00000000000000
------------------------------------------------------------
The following tests failed: _test_root_lattice_realization
"""
if not hasattr(self.cartan_type(),"AmbientSpace"):
return None
AmbientSpace = self.cartan_type().AmbientSpace
if not base_ring.has_coerce_map_from(AmbientSpace.smallest_base_ring(self.cartan_type())):
return None
return AmbientSpace(self, base_ring)
class RootSystem(UniqueRepresentation, SageObject):
r"""
A class for root systems.
EXAMPLES:
We construct the root system for type `B_3`::
sage: R=RootSystem(['B',3]); R
Root system of type ['B', 3]
``R`` models the root system abstractly. It comes equipped with various
realizations of the root and weight lattices, where all computations
take place. Let us play first with the root lattice::
sage: space = R.root_lattice()
sage: space
Root lattice of the Root system of type ['B', 3]
It is the free `\ZZ`-module
`\bigoplus_i \ZZ.\alpha_i` spanned by the simple
roots::
sage: space.base_ring()
Integer Ring
sage: list(space.basis())
[alpha[1], alpha[2], alpha[3]]
Let us do some computations with the simple roots::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
alpha[1] + alpha[2]
There is a canonical pairing between the root lattice and the
coroot lattice::
sage: R.coroot_lattice()
Coroot lattice of the Root system of type ['B', 3]
We construct the simple coroots, and do some computations (see
comments about duality below for some caveat)::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
We can carry over the same computations in any of the other
realizations of the root lattice, like the root space
`\bigoplus_i \QQ.\alpha_i`, the weight lattice
`\bigoplus_i \ZZ.\Lambda_i`, the weight
space `\bigoplus_i \QQ.\Lambda_i`. For example::
sage: space = R.weight_space()
sage: space
Weight space over the Rational Field of the Root system of type ['B', 3]
::
sage: space.base_ring()
Rational Field
sage: list(space.basis())
[Lambda[1], Lambda[2], Lambda[3]]
::
sage: alpha = space.simple_roots()
sage: alpha[1] + alpha[2]
Lambda[1] + Lambda[2] - 2*Lambda[3]
The fundamental weights are the dual basis of the coroots::
sage: Lambda = space.fundamental_weights()
sage: Lambda[1]
Lambda[1]
::
sage: alphacheck = space.simple_coroots()
sage: list(alphacheck)
[alphacheck[1], alphacheck[2], alphacheck[3]]
::
sage: [Lambda[i].scalar(alphacheck[1]) for i in space.index_set()]
[1, 0, 0]
sage: [Lambda[i].scalar(alphacheck[2]) for i in space.index_set()]
[0, 1, 0]
sage: [Lambda[i].scalar(alphacheck[3]) for i in space.index_set()]
[0, 0, 1]
Let us use the simple reflections. In the weight space, they
work as in the *number game*: firing the node `i` on an
element `x` adds `c` times the simple root
`\alpha_i`, where `c` is the coefficient of
`i` in `x`::
sage: s = space.simple_reflections()
sage: Lambda[1].simple_reflection(1)
-Lambda[1] + Lambda[2]
sage: Lambda[2].simple_reflection(1)
Lambda[2]
sage: Lambda[3].simple_reflection(1)
Lambda[3]
sage: (-2*Lambda[1] + Lambda[2] + Lambda[3]).simple_reflection(1)
2*Lambda[1] - Lambda[2] + Lambda[3]
It can be convenient to manipulate the simple reflections
themselves::
sage: s = space.simple_reflections()
sage: s[1](Lambda[1])
-Lambda[1] + Lambda[2]
sage: s[1](Lambda[2])
Lambda[2]
sage: s[1](Lambda[3])
Lambda[3]
The root system may also come equipped with an ambient space, that
is a simultaneous realization of the weight lattice and the coroot
lattice in a Euclidean vector space. This is implemented on a type
by type basis, and is not always available. When the coefficients
permit it, this is also available as an ambient lattice.
TODO: Demo: signed permutations realization of type B
The root system is aware of its dual root system::
sage: R.dual
Dual of root system of type ['B', 3]
R.dual is really the root system of type `C_3`::
sage: R.dual.cartan_type()
['C', 3]
And the coroot lattice that we have been manipulating before is
really implemented as the root lattice of the dual root system::
sage: R.dual.root_lattice()
Coroot lattice of the Root system of type ['B', 3]
In particular, the coroots for the root lattice are in fact the
roots of the coroot lattice::
sage: list(R.root_lattice().simple_coroots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.coroot_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
sage: list(R.dual.root_lattice().simple_roots())
[alphacheck[1], alphacheck[2], alphacheck[3]]
The coweight lattice and space are defined similarly. Note that, to
limit confusion, all the output have been tweaked appropriately.
.. seealso::
- :mod:`sage.combinat.root_system`
- :class:`RootSpace`
- :class:`WeightSpace`
- :class:`AmbientSpace`
- :class:`~sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations`
- :class:`~sage.combinat.root_system.weight_lattice_realizations.WeightLatticeRealizations`
TESTS::
sage: R = RootSystem(['C',3])
sage: TestSuite(R).run()
sage: L = R.ambient_space()
sage: s = L.simple_reflections() # this used to break the testsuite below due to caching an unpicklable method
sage: s = L.simple_projections() # todo: not implemented
sage: TestSuite(L).run()
sage: L = R.root_space()
sage: s = L.simple_reflections()
sage: TestSuite(L).run()
::
sage: for T in CartanType.samples(crystalographic=True): # long time (13s on sage.math, 2012)
... TestSuite(RootSystem(T)).run()
"""
@staticmethod
def __classcall__(cls, cartan_type, as_dual_of=None):
"""
Straighten arguments to enable unique representation
.. seealso:: :class:`UniqueRepresentation`
TESTS::
sage: RootSystem(["A",3]) is RootSystem(CartanType(["A",3]))
True
sage: RootSystem(["B",3], as_dual_of=None) is RootSystem("B3")
True
"""
return super(RootSystem, cls).__classcall__(cls, CartanType(cartan_type), as_dual_of)
def __init__(self, cartan_type, as_dual_of=None):
"""
TESTS::
sage: R = RootSystem(['A',3])
sage: R
Root system of type ['A', 3]
"""
self._cartan_type = CartanType(cartan_type)
# Duality
# The root system can be defined as dual of another root system. This will
# only affects the pretty printing
if as_dual_of is None:
self.dual_side = False
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
# still fails for CartanType G2xA1
try:
self.dual = RootSystem(self._cartan_type.dual(), as_dual_of=self);
except:
pass
else:
self.dual_side = True
self.dual = as_dual_of
def _test_root_lattice_realizations(self, **options):
"""
Runs tests on all the root lattice realizations of this root
system.
EXAMPLES::
sage: RootSystem(["A",3])._test_root_lattice_realizations()
See also :class:`TestSuite`.
"""
tester = self._tester(**options)
options.pop('tester', None)
from sage.misc.sage_unittest import TestSuite
TestSuite(self.root_lattice()).run(**options)
TestSuite(self.root_space()).run(**options)
TestSuite(self.weight_lattice()).run(**options)
TestSuite(self.weight_space()).run(**options)
if self.cartan_type().is_affine():
TestSuite(self.weight_lattice(extended=True)).run(**options)
TestSuite(self.weight_space(extended=True)).run(**options)
if self.ambient_lattice() is not None:
TestSuite(self.ambient_lattice()).run(**options)
if self.ambient_space() is not None:
TestSuite(self.ambient_space()).run(**options)
def _repr_(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]) # indirect doctest
Root system of type ['A', 3]
sage: RootSystem(['B',3]).dual # indirect doctest
Dual of root system of type ['B', 3]
"""
if self.dual_side:
return "Dual of root system of type %s"%self.dual.cartan_type()
else:
return "Root system of type %s"%self.cartan_type()
def cartan_type(self):
"""
Returns the Cartan type of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.cartan_type()
['A', 3]
"""
return self._cartan_type
@cached_method
def dynkin_diagram(self):
"""
Returns the Dynkin diagram of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3])
sage: R.dynkin_diagram()
O---O---O
1 2 3
A3
"""
return self.cartan_type().dynkin_diagram()
@cached_method
def cartan_matrix(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).cartan_matrix()
[ 2 -1 0]
[-1 2 -1]
[ 0 -1 2]
"""
return self.cartan_type().cartan_matrix()
def index_set(self):
"""
EXAMPLES::
sage: RootSystem(['A',3]).index_set()
[1, 2, 3]
"""
return self.cartan_type().index_set()
@cached_method
def is_finite(self):
"""
Returns True if self is a finite root system.
EXAMPLES::
sage: RootSystem(["A",3]).is_finite()
True
sage: RootSystem(["A",3,1]).is_finite()
False
"""
return self.cartan_type().is_finite()
@cached_method
def is_irreducible(self):
"""
Returns True if self is an irreducible root system.
EXAMPLES::
sage: RootSystem(['A', 3]).is_irreducible()
True
sage: RootSystem("A2xB2").is_irreducible()
False
"""
return self.cartan_type().is_irreducible()
def __cmp__(self, other):
"""
EXAMPLES::
sage: r1 = RootSystem(['A',3])
sage: r2 = RootSystem(['B',3])
sage: r1 == r1
True
sage: r1 == r2
False
"""
if self.__class__ != other.__class__:
return cmp(self.__class__, other.__class__)
if self._cartan_type != other._cartan_type:
return cmp(self._cartan_type, other._cartan_type)
return 0
def root_lattice(self):
"""
Returns the root lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_lattice()
Root lattice of the Root system of type ['A', 3]
"""
return self.root_space(ZZ)
@cached_method
def root_space(self, base_ring=QQ):
"""
Returns the root space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).root_space()
Root space over the Rational Field of the Root system of type ['A', 3]
"""
return RootSpace(self, base_ring)
def root_poset(self, restricted=False, facade=False):
r"""
Returns the (restricted) root poset associated to ``self``.
The elements are given by the positive roots (resp. non-simple, positive roots), and
`\alpha \leq \beta` iff `\beta - \alpha` is a non-negative linear combination of simple roots.
INPUT:
- ``restricted`` -- (default:False) if True, only non-simple roots are considered.
- ``facade`` -- (default:False) passes facade option to the poset generator.
EXAMPLES::
sage: Phi = RootSystem(['A',2]).root_poset(); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]]]
sage: Phi = RootSystem(['A',3]).root_poset(restricted=True); Phi
Finite poset containing 3 elements
sage: Phi.cover_relations()
[[alpha[1] + alpha[2], alpha[1] + alpha[2] + alpha[3]], [alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
sage: Phi = RootSystem(['B',2]).root_poset(); Phi
Finite poset containing 4 elements
sage: Phi.cover_relations()
[[alpha[1], alpha[1] + alpha[2]], [alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2], alpha[1] + 2*alpha[2]]]
"""
return self.root_lattice().root_poset(restricted=restricted,facade=facade)
def coroot_lattice(self):
"""
Returns the coroot lattice associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_lattice()
Coroot lattice of the Root system of type ['A', 3]
"""
return self.dual.root_lattice()
def coroot_space(self, base_ring=QQ):
"""
Returns the coroot space associated to self.
EXAMPLES::
sage: RootSystem(['A',3]).coroot_space()
Coroot space over the Rational Field of the Root system of type ['A', 3]
"""
return self.dual.root_space(base_ring)
@cached_method
def weight_lattice(self, extended = False):
"""
Returns the weight lattice associated to self.
.. see also::
- :meth:`weight_space`
- :meth:`coweight_space`, :meth:`coweight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).weight_lattice()
Weight lattice of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).weight_space(extended = True)
Extended weight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return WeightSpace(self, ZZ, extended = extended)
@cached_method
def weight_space(self, base_ring=QQ, extended = False):
"""
Returns the weight space associated to self.
.. see also::
- :meth:`weight_lattice`
- :meth:`coweight_space`, :meth:`coweight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).weight_space()
Weight space over the Rational Field of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).weight_space(extended = True)
Extended weight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return WeightSpace(self, base_ring, extended = extended)
def coweight_lattice(self, extended = False):
"""
Returns the coweight lattice associated to self.
This is the weight lattice of the dual root system.
.. see also::
- :meth:`coweight_space`
- :meth:`weight_space`, :meth:`weight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).coweight_lattice()
Coweight lattice of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).coweight_lattice(extended = True)
Extended coweight lattice of the Root system of type ['A', 3, 1]
"""
return self.dual.weight_lattice(extended = extended)
def coweight_space(self, base_ring=QQ, extended = False):
"""
Returns the coweight space associated to self.
This is the weight space of the dual root system.
.. see also::
- :meth:`coweight_lattice`
- :meth:`weight_space`, :meth:`weight_lattice`
- :class:`~sage.combinat.root_system.WeightSpace`
EXAMPLES::
sage: RootSystem(['A',3]).coweight_space()
Coweight space over the Rational Field of the Root system of type ['A', 3]
sage: RootSystem(['A',3,1]).coweight_space(extended=True)
Extended coweight space over the Rational Field of the Root system of type ['A', 3, 1]
"""
return self.dual.weight_space(base_ring, extended = extended)
def ambient_lattice(self):
r"""
Returns the usual ambient lattice for this root_system, if it
exists and is implemented, and None otherwise. This is a
`\ZZ`-module, endowed with its canonical euclidean
scalar product, which embeds simultaneously the root lattice
and the coroot lattice (what about the weight lattice?)
EXAMPLES::
sage: RootSystem(['A',4]).ambient_lattice()
Ambient lattice of the Root system of type ['A', 4]
::
sage: RootSystem(['B',4]).ambient_lattice()
sage: RootSystem(['C',4]).ambient_lattice()
sage: RootSystem(['D',4]).ambient_lattice()
sage: RootSystem(['E',6]).ambient_lattice()
sage: RootSystem(['F',4]).ambient_lattice()
sage: RootSystem(['G',2]).ambient_lattice()
"""
return self.ambient_space(ZZ)
@cached_method
def ambient_space(self, base_ring=QQ):
r"""
Returns the usual ambient space for this root_system, if it is
implemented, and None otherwise. This is a `\QQ`-module, endowed with
its canonical euclidean scalar product, which embeds simultaneously
the root lattice and the coroot lattice (what about the weight
lattice?). An alternative base ring can be provided as an option;
it must contain the smallest ring over which the ambient space can
be defined (`\ZZ` or `\QQ`, depending on the type).
EXAMPLES::
sage: RootSystem(['A',4]).ambient_space()
Ambient space of the Root system of type ['A', 4]
::
sage: RootSystem(['B',4]).ambient_space()
Ambient space of the Root system of type ['B', 4]
::
sage: RootSystem(['C',4]).ambient_space()
Ambient space of the Root system of type ['C', 4]
::
sage: RootSystem(['D',4]).ambient_space()
Ambient space of the Root system of type ['D', 4]
::
sage: RootSystem(['E',6]).ambient_space()
Ambient space of the Root system of type ['E', 6]
::
sage: RootSystem(['F',4]).ambient_space()
Ambient space of the Root system of type ['F', 4]
::
sage: RootSystem(['G',2]).ambient_space()
Ambient space of the Root system of type ['G', 2]
"""
# Intention: check that the ambient_space is implemented and that
# base_ring contains the smallest base ring for this ambient space
if not hasattr(self.cartan_type(),"AmbientSpace"):
return None
AmbientSpace = self.cartan_type().AmbientSpace
if base_ring == ZZ and AmbientSpace.smallest_base_ring() == QQ:
return None
return AmbientSpace(self, base_ring)