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r""" Root systems ============
See :ref:`sage.combinat.root_system` for an overview. """ from __future__ import absolute_import #***************************************************************************** # Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>, # Justin Walker <justin at mac.com> # 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #***************************************************************************** # Design largely inspired from MuPAD-Combinat from sage.structure.sage_object import SageObject from sage.structure.unique_representation import UniqueRepresentation from .cartan_type import CartanType from sage.rings.all import ZZ, QQ from sage.misc.all import cached_method from .root_space import RootSpace from .weight_space import WeightSpace
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 crystallographic 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), (3, 2, 1), (3, 1, 2), (2, 1, 3), (2, 3, 1), (1, 3, 2)]
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), (-2, 2, 6), (-1, 1, 6), (0, 1, 5), (-2, 3, 5), (-1, 3, 4)]
We can also plot various components of the ambient spaces::
sage: L = RootSystem(['A',2]).ambient_space() sage: L.plot() Graphics object consisting of 13 graphics primitives
For more on plotting, see :ref:`sage.combinat.root_system.plot`.
.. 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(crystallographic=True): # long time (13s on sage.math, 2012) ....: TestSuite(RootSystem(T)).run()
Some checks for equality::
sage: r1 = RootSystem(['A',3]) sage: r2 = RootSystem(['B',3]) sage: r1 == r1 True sage: r1 == r2 False sage: r1 != r1 False
Check that root systems inherit a hash method from ``UniqueRepresentation``::
sage: hash(r1) # random 42 """
@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 """
def __init__(self, cartan_type, as_dual_of=None): """ TESTS::
sage: R = RootSystem(['A',3]) sage: R Root system of type ['A', 3] """
# Duality # The root system can be defined as dual of another root system. This will # only affects the pretty printing # still fails for CartanType G2xA1 else:
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()
.. SEEALSO:: :class:`TestSuite`. """ TestSuite(self.weight_lattice(extended=True)).run(**options) TestSuite(self.weight_space(extended=True)).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] """ else:
def cartan_type(self): """ Returns the Cartan type of the root system.
EXAMPLES::
sage: R = RootSystem(['A',3]) sage: R.cartan_type() ['A', 3] """
@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 """
@cached_method def cartan_matrix(self): """ EXAMPLES::
sage: RootSystem(['A',3]).cartan_matrix() [ 2 -1 0] [-1 2 -1] [ 0 -1 2] """
@cached_method def index_set(self): """ EXAMPLES::
sage: RootSystem(['A',3]).index_set() (1, 2, 3) """
@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 """
@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 """
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] """
@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] """
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: sorted(Phi.cover_relations(), key=str) [[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: sorted(Phi.cover_relations(), key=str) [[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[2], alpha[1] + alpha[2]], [alpha[1], alpha[1] + alpha[2]], [alpha[1] + alpha[2], alpha[1] + 2*alpha[2]]] """
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] """
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] """
@cached_method def weight_lattice(self, extended = False): """ Returns the weight lattice associated to self.
.. SEEALSO::
- :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] """
@cached_method def weight_space(self, base_ring=QQ, extended = False): """ Returns the weight space associated to self.
.. SEEALSO::
- :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] """
def coweight_lattice(self, extended = False): """ Returns the coweight lattice associated to self.
This is the weight lattice of the dual root system.
.. SEEALSO::
- :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] """
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.
.. SEEALSO::
- :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] """
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() """
@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 it is constructed from hard coded type by type data, according to the usual Bourbaki conventions. Such data is provided for all the finite (crystallographic) 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 """
def coambient_space(self, base_ring=QQ): r""" Return the coambient space for this root system.
This is the ambient space of the dual root system.
.. SEEALSO::
- :meth:`ambient_space`
EXAMPLES::
sage: L = RootSystem(["B",2]).ambient_space(); L Ambient space of the Root system of type ['B', 2] sage: coL = RootSystem(["B",2]).coambient_space(); coL Coambient space of the Root system of type ['B', 2]
The roots and coroots are interchanged::
sage: coL.simple_roots() Finite family {1: (1, -1), 2: (0, 2)} sage: L.simple_coroots() Finite family {1: (1, -1), 2: (0, 2)}
sage: coL.simple_coroots() Finite family {1: (1, -1), 2: (0, 1)} sage: L.simple_roots() Finite family {1: (1, -1), 2: (0, 1)} """
def WeylDim(ct, coeffs): """ The Weyl Dimension Formula.
INPUT:
- ``type`` - a Cartan type
- ``coeffs`` - a list of nonnegative integers
The length of the list must equal the rank type[1]. A dominant weight hwv is constructed by summing the fundamental weights with coefficients from this list. The dimension of the irreducible representation of the semisimple complex Lie algebra with highest weight vector hwv is returned.
EXAMPLES:
For `SO(7)`, the Cartan type is `B_3`, so::
sage: WeylDim(['B',3],[1,0,0]) # standard representation of SO(7) 7 sage: WeylDim(['B',3],[0,1,0]) # exterior square 21 sage: WeylDim(['B',3],[0,0,1]) # spin representation of spin(7) 8 sage: WeylDim(['B',3],[1,0,1]) # sum of the first and third fundamental weights 48 sage: [WeylDim(['F',4],x) for x in [1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] [52, 1274, 273, 26] sage: [WeylDim(['E', 6], x) for x in [0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 2], [0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1], [2, 0, 0, 0, 0, 0]] [1, 78, 27, 351, 351, 351, 27, 650, 351] """ |