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r""" Classical Crystals """ #***************************************************************************** # Copyright (C) 2010 Anne Schilling <anne at math.ucdavis.edu> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #******************************************************************************
from sage.misc.cachefunc import cached_method from sage.categories.category_singleton import Category_singleton from sage.categories.crystals import Crystals from sage.categories.finite_crystals import FiniteCrystals from sage.categories.regular_crystals import RegularCrystals from sage.categories.highest_weight_crystals import HighestWeightCrystals from sage.categories.tensor import TensorProductsCategory
class ClassicalCrystals(Category_singleton): """ The category of classical crystals, that is crystals of finite Cartan type.
EXAMPLES::
sage: C = ClassicalCrystals() sage: C Category of classical crystals sage: C.super_categories() [Category of regular crystals, Category of finite crystals, Category of highest weight crystals] sage: C.example() Highest weight crystal of type A_3 of highest weight omega_1
TESTS::
sage: TestSuite(C).run() sage: B = ClassicalCrystals().example() sage: TestSuite(B).run(verbose = True) running ._test_an_element() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_stembridge_local_axioms() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . pass running ._test_eq() . . . pass running ._test_fast_iter() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_some_elements() . . . pass running ._test_stembridge_local_axioms() . . . pass """
def super_categories(self): r""" EXAMPLES::
sage: ClassicalCrystals().super_categories() [Category of regular crystals, Category of finite crystals, Category of highest weight crystals] """
def example(self, n = 3): """ Returns an example of highest weight crystals, as per :meth:`Category.example`.
EXAMPLES::
sage: B = ClassicalCrystals().example(); B Highest weight crystal of type A_3 of highest weight omega_1 """
def additional_structure(self): r""" Return ``None``.
Indeed, the category of classical crystals defines no additional structure: it only states that its objects are `U_q(\mathfrak{g})`-crystals, where `\mathfrak{g}` is of finite type.
.. SEEALSO:: :meth:`Category.additional_structure`
EXAMPLES::
sage: ClassicalCrystals().additional_structure() """
class ParentMethods:
def opposition_automorphism(self): r""" Deprecated in :trac:`15560`. Use the corresponding method in Cartan type.
EXAMPLES::
sage: T = crystals.Tableaux(['A',5],shape=[1]) sage: T.opposition_automorphism() doctest:...: DeprecationWarning: opposition_automorphism is deprecated. Use opposition_automorphism from the Cartan type instead. See http://trac.sagemath.org/15560 for details. Finite family {1: 5, 2: 4, 3: 3, 4: 2, 5: 1} """ ' opposition_automorphism from the Cartan type instead.')
def demazure_character(self, w, f = None): r""" Returns the Demazure character associated to ``w``.
INPUT:
- ``w`` -- an element of the ambient weight lattice realization of the crystal, or a reduced word, or an element in the associated Weyl group
OPTIONAL:
- ``f`` -- a function from the crystal to a module
This is currently only supported for crystals whose underlying weight space is the ambient space.
The Demazure character is obtained by applying the Demazure operator `D_w` (see :meth:`sage.categories.regular_crystals.RegularCrystals.ParentMethods.demazure_operator`) to the highest weight element of the classical crystal. The simple Demazure operators `D_i` (see :meth:`sage.categories.regular_crystals.RegularCrystals.ElementMethods.demazure_operator_simple`) do not braid on the level of crystals, but on the level of characters they do. That is why it makes sense to input ``w`` either as a weight, a reduced word, or as an element of the underlying Weyl group.
EXAMPLES::
sage: T = crystals.Tableaux(['A',2], shape = [2,1]) sage: e = T.weight_lattice_realization().basis() sage: weight = e[0] + 2*e[2] sage: weight.reduced_word() [2, 1] sage: T.demazure_character(weight) x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x1*x3^2
sage: T = crystals.Tableaux(['A',3],shape=[2,1]) sage: T.demazure_character([1,2,3]) x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3 sage: W = WeylGroup(['A',3]) sage: w = W.from_reduced_word([1,2,3]) sage: T.demazure_character(w) x1^2*x2 + x1*x2^2 + x1^2*x3 + x1*x2*x3 + x2^2*x3
sage: T = crystals.Tableaux(['B',2], shape = [2]) sage: e = T.weight_lattice_realization().basis() sage: weight = -2*e[1] sage: T.demazure_character(weight) x1^2 + x1*x2 + x2^2 + x1 + x2 + x1/x2 + 1/x2 + 1/x2^2 + 1
sage: T = crystals.Tableaux("B2",shape=[1/2,1/2]) sage: b2=WeylCharacterRing("B2",base_ring=QQ).ambient() sage: T.demazure_character([1,2],f=lambda x:b2(x.weight())) b2(-1/2,1/2) + b2(1/2,-1/2) + b2(1/2,1/2)
REFERENCES:
- [De1974]_
- [Ma2009]_ """ else: # TODO: use P.linear_combination when PolynomialRing will be a ModulesWithBasis else:
def character(self, R=None): """ Returns the character of this crystal.
INPUT:
- ``R`` -- a :class:`WeylCharacterRing` (default: the default :class:`WeylCharacterRing` for this Cartan type)
Returns the character of ``self`` as an element of ``R``.
EXAMPLES::
sage: C = crystals.Tableaux("A2", shape=[2,1]) sage: chi = C.character(); chi A2(2,1,0)
sage: T = crystals.TensorProduct(C,C) sage: chiT = T.character(); chiT A2(2,2,2) + 2*A2(3,2,1) + A2(3,3,0) + A2(4,1,1) + A2(4,2,0) sage: chiT == chi^2 True
One may specify an alternate :class:`WeylCharacterRing`::
sage: R = WeylCharacterRing("A2", style="coroots") sage: chiT = T.character(R); chiT A2(0,0) + 2*A2(1,1) + A2(0,3) + A2(3,0) + A2(2,2) sage: chiT in R True
It should have the same Cartan type and use the same realization of the weight lattice as ``self``::
sage: R = WeylCharacterRing("A3", style="coroots") sage: T.character(R) Traceback (most recent call last): ... ValueError: Weyl character ring does not have the right Cartan type
"""
def __iter__(self): r""" Returns an iterator over the elements of this crystal.
This iterator uses little memory, storing only one element of the crystal at a time. For details on the complexity, see :class:`sage.combinat.crystals.crystals.CrystalBacktracker`.
EXAMPLES::
sage: C = crystals.Letters(['A',5]) sage: [x for x in C] [1, 2, 3, 4, 5, 6]
TESTS::
sage: C = crystals.Letters(['D',4]) sage: D = crystals.SpinsPlus(['D',4]) sage: E = crystals.SpinsMinus(['D',4]) sage: T = crystals.TensorProduct(D,E,generators=[[D.list()[0],E.list()[0]]]) sage: U = crystals.TensorProduct(C,E,generators=[[C(1),E.list()[0]]]) sage: T.cardinality() 56
sage: TestSuite(T).run(verbose = True) running ._test_an_element() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_stembridge_local_axioms() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . pass running ._test_eq() . . . pass running ._test_fast_iter() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_some_elements() . . . pass running ._test_stembridge_local_axioms() . . . pass
sage: TestSuite(U).run(verbose = True) running ._test_an_element() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_stembridge_local_axioms() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . pass running ._test_eq() . . . pass running ._test_fast_iter() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_some_elements() . . . pass running ._test_stembridge_local_axioms() . . . pass
Bump's systematic tests::
sage: fa3 = lambda a,b,c: crystals.Tableaux(['A',3],shape=[a+b+c,b+c,c]) sage: fb3 = lambda a,b,c: crystals.Tableaux(['B',3],shape=[a+b+c,b+c,c]) sage: fc3 = lambda a,b,c: crystals.Tableaux(['C',3],shape=[a+b+c,b+c,c]) sage: fb4 = lambda a,b,c,d: crystals.Tableaux(['B',4],shape=[a+b+c+d,b+c+d,c+d,d]) sage: fd4 = lambda a,b,c,d: crystals.Tableaux(['D',4],shape=[a+b+c+d,b+c+d,c+d,d]) sage: fd5 = lambda a,b,c,d,e: crystals.Tableaux(['D',5],shape=[a+b+c+d+e,b+c+d+e,c+d+e,d+e,e]) sage: def fd4spinplus(a,b,c,d): ....: C = crystals.Tableaux(['D',4],shape=[a+b+c+d,b+c+d,c+d,d]) ....: D = crystals.SpinsPlus(['D',4]) ....: return crystals.TensorProduct(C,D,generators=[[C[0],D[0]]]) sage: def fb3spin(a,b,c): ....: C = crystals.Tableaux(['B',3],shape=[a+b+c,b+c,c]) ....: D = crystals.Spins(['B',3]) ....: return crystals.TensorProduct(C,D,generators=[[C[0],D[0]]])
.. TODO::
Choose a good panel of values for `a,b,c ...` both for basic systematic tests and for conditionally run, computationally involved tests.
::
sage: TestSuite(fb4(1,0,1,0)).run(verbose = True) # long time (8s on sage.math, 2011) running ._test_an_element() . . . pass running ._test_cardinality() . . . pass running ._test_category() . . . pass running ._test_elements() . . . Running the test suite of self.an_element() running ._test_category() . . . pass running ._test_eq() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_stembridge_local_axioms() . . . pass pass running ._test_elements_eq_reflexive() . . . pass running ._test_elements_eq_symmetric() . . . pass running ._test_elements_eq_transitive() . . . pass running ._test_elements_neq() . . . pass running ._test_enumerated_set_contains() . . . pass running ._test_enumerated_set_iter_cardinality() . . . pass running ._test_enumerated_set_iter_list() . . . pass running ._test_eq() . . . pass running ._test_fast_iter() . . . pass running ._test_new() . . . pass running ._test_not_implemented_methods() . . . pass running ._test_pickling() . . . pass running ._test_some_elements() . . . pass running ._test_stembridge_local_axioms() . . . pass
::
#sage: fb4(1,1,1,1).check() # expensive: the crystal is of size 297297 #True """
def _test_fast_iter(self, **options): r""" Tests whether the elements returned by :meth:`.__iter__` and ``Crystal.list(self)`` are the same (the two algorithms are different).
EXAMPLES::
sage: C = crystals.Letters(['A', 5]) sage: C._test_fast_iter() """
def cardinality(self): r""" Returns the number of elements of the crystal, using Weyl's dimension formula on each connected component.
EXAMPLES::
sage: C = ClassicalCrystals().example(5) sage: C.cardinality() 6 """ for x in self.highest_weight_vectors())
class ElementMethods:
def lusztig_involution(self): r""" Return the Lusztig involution on the classical highest weight crystal ``self``.
The Lusztig involution on a finite-dimensional highest weight crystal `B(\lambda)` of highest weight `\lambda` maps the highest weight vector to the lowest weight vector and the Kashiwara operator `f_i` to `e_{i^*}`, where `i^*` is defined as `\alpha_{i^*} = -w_0(\alpha_i)`. Here `w_0` is the longest element of the Weyl group acting on the `i`-th simple root `\alpha_i`.
EXAMPLES::
sage: B = crystals.Tableaux(['A',3],shape=[2,1]) sage: b = B(rows=[[1,2],[4]]) sage: b.lusztig_involution() [[1, 4], [3]] sage: b.to_tableau().schuetzenberger_involution(n=4) [[1, 4], [3]]
sage: all(b.lusztig_involution().to_tableau() == b.to_tableau().schuetzenberger_involution(n=4) for b in B) True
sage: B = crystals.Tableaux(['D',4],shape=[1]) sage: [[b,b.lusztig_involution()] for b in B] [[[[1]], [[-1]]], [[[2]], [[-2]]], [[[3]], [[-3]]], [[[4]], [[-4]]], [[[-4]], [[4]]], [[[-3]], [[3]]], [[[-2]], [[2]]], [[[-1]], [[1]]]]
sage: B = crystals.Tableaux(['D',3],shape=[1]) sage: [[b,b.lusztig_involution()] for b in B] [[[[1]], [[-1]]], [[[2]], [[-2]]], [[[3]], [[3]]], [[[-3]], [[-3]]], [[[-2]], [[2]]], [[[-1]], [[1]]]]
sage: C = CartanType(['E',6]) sage: La = C.root_system().weight_lattice().fundamental_weights() sage: T = crystals.HighestWeight(La[1]) sage: t = T[3]; t [(-4, 2, 5)] sage: t.lusztig_involution() [(-2, -3, 4)] """
class TensorProducts(TensorProductsCategory): """ The category of classical crystals constructed by tensor product of classical crystals. """ @cached_method def extra_super_categories(self): """ EXAMPLES::
sage: ClassicalCrystals().TensorProducts().extra_super_categories() [Category of classical crystals] """
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