CoCalc -- highest_weight

GitHub Repository: sagemath/sagelib
Path: blob/master/sage/combinat/crystals/highest_weight_crystals.py
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r"""
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Highest weight crystals
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"""
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#*****************************************************************************
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#       Copyright (C) 2009   Anne Schilling <anne at math.ucdavis.edu>
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#
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#  Distributed under the terms of the GNU General Public License (GPL)
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#
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#    This code is distributed in the hope that it will be useful,
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#    but WITHOUT ANY WARRANTY; without even the implied warranty of
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#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
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#    General Public License for more details.
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#
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#  The full text of the GPL is available at:
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#
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#                  http://www.gnu.org/licenses/
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#****************************************************************************
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from sage.categories.classical_crystals import ClassicalCrystals
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from sage.structure.parent import Parent
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from sage.combinat.root_system.cartan_type import CartanType
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from sage.combinat.crystals.letters import CrystalOfLetters
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from sage.combinat.crystals.tensor_product import TensorProductOfCrystals, CrystalOfWords
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def HighestWeightCrystal(dominant_weight):
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    r"""
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    Returns an implementation of the highest weight crystal of highest weight `dominant_weight`.
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    This is currently only implemented for crystals of type `E_6` and `E_7`.
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    TODO: implement highest weight crystals for classical types `A_n`, `B_n`, `C_n`, `D_n` using tableaux.
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    TODO: implement the Littelmann path model or alcove model to obtain a realization
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    for any highest weight crystal of given type (even affine). 
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    EXAMPLES::
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        sage: C=CartanType(['E',6])
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        sage: La=C.root_system().weight_lattice().fundamental_weights()
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        sage: T = HighestWeightCrystal(La[1])
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        sage: T.cardinality()
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        27
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        sage: T = HighestWeightCrystal(La[6])
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        sage: T.cardinality()
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        27
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        sage: T = HighestWeightCrystal(La[2])
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        sage: T.cardinality()
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        78
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        sage: T = HighestWeightCrystal(La[4])
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        sage: T.cardinality()
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        2925
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        sage: T = HighestWeightCrystal(La[3])
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        sage: T.cardinality()
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        351
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        sage: T = HighestWeightCrystal(La[5])
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        sage: T.cardinality()
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        351
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        sage: C=CartanType(['E',7])
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        sage: La=C.root_system().weight_lattice().fundamental_weights()
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        sage: T = HighestWeightCrystal(La[1])
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        sage: T.cardinality()
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        133
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        sage: T = HighestWeightCrystal(La[2])
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        sage: T.cardinality()
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        912
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        sage: T = HighestWeightCrystal(La[3])
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        sage: T.cardinality()
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        8645
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        sage: T = HighestWeightCrystal(La[4])
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        sage: T.cardinality()
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        365750
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        sage: T = HighestWeightCrystal(La[5])
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        sage: T.cardinality()
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        27664
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        sage: T = HighestWeightCrystal(La[6])
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        sage: T.cardinality()
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        1539
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        sage: T = HighestWeightCrystal(La[7])
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        sage: T.cardinality()
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        56
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    """
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    cartan_type = dominant_weight.parent().cartan_type()
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    if cartan_type.is_finite() and cartan_type.type() in ['A','B','C','D']:
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        raise NotImplementedError
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    elif cartan_type == CartanType(['E',6]):
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        return FiniteDimensionalHighestWeightCrystal_TypeE6(dominant_weight)
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    elif cartan_type == CartanType(['E',7]):
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        return FiniteDimensionalHighestWeightCrystal_TypeE7(dominant_weight)
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    else:
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        raise NotImplementedError
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class FiniteDimensionalHighestWeightCrystal_TypeE(CrystalOfWords):
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    """
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    Commonalities for all finite dimensional type E highest weight crystals
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    Subclasses should setup an attribute column_crystal in their
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    __init__ method before calling the __init__ method of this class.
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    """
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    def __init__(self, dominant_weight):
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        """
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        EXAMPLES::
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            sage: C=CartanType(['E',6])
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            sage: La=C.root_system().weight_lattice().fundamental_weights()
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            sage: T = HighestWeightCrystal(2*La[2])
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            sage: T.cartan_type()
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            ['E', 6]
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            sage: T.module_generators
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            [[[[2, -1], [1]], [[2, -1], [1]]]]
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            sage: T.cardinality()
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            2430
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            sage: T = HighestWeightCrystal(La[2])
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            sage: T.cardinality()
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            78
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        """
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        self._cartan_type = dominant_weight.parent().cartan_type()
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        self._highest_weight = dominant_weight
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        assert dominant_weight.is_dominant()
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        self.rename("Finite dimensional highest weight crystal of type %s and highest weight %s"%(self._cartan_type, dominant_weight))
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        Parent.__init__(self, category = ClassicalCrystals())
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        self.module_generators = [self.module_generator()]
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    def module_generator(self):
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        """
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        This yields the module generator (or highest weight element) of the classical
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        crystal of given dominant weight in self.
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        EXAMPLES::
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            sage: C=CartanType(['E',6])
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            sage: La=C.root_system().weight_lattice().fundamental_weights()
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            sage: T = HighestWeightCrystal(La[2])
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            sage: T.module_generator()
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            [[[2, -1], [1]]]
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            sage: T = HighestWeightCrystal(0*La[2])
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            sage: T.module_generator()
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            []
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            sage: C=CartanType(['E',7])
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            sage: La=C.root_system().weight_lattice().fundamental_weights()
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            sage: T = HighestWeightCrystal(La[1])
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            sage: T.module_generator()
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            [[[-7, 1], [7]]]
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        """
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        dominant_weight = self._highest_weight
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        tensor = sum(( [self.column_crystal[i]]*dominant_weight.coefficient(i) for i in dominant_weight.support()), [])
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        return self._element_constructor_(*[B.module_generators[0] for B in tensor])
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class FiniteDimensionalHighestWeightCrystal_TypeE6(FiniteDimensionalHighestWeightCrystal_TypeE):
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    r"""
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    Class of finite dimensional highest weight crystals of type `E_6`.
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    EXAMPLES::
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        sage: C=CartanType(['E',6])
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        sage: La=C.root_system().weight_lattice().fundamental_weights()
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        sage: T = HighestWeightCrystal(La[2]); T
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        Finite dimensional highest weight crystal of type ['E', 6] and highest weight Lambda[2]
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        sage: B1 = T.column_crystal[1]; B1
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        The crystal of letters for type ['E', 6]
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        sage: B6 = T.column_crystal[6]; B6
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        The crystal of letters for type ['E', 6] (dual)
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        sage: t = T(B6([-1]),B1([-1,3])); t
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        [[-1], [-1, 3]]
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        sage: [t.epsilon(i) for i in T.index_set()]
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        [2, 0, 0, 0, 0, 0]
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        sage: [t.phi(i) for i in T.index_set()]
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        [0, 0, 1, 0, 0, 0]
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        sage: TestSuite(t).run()
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    """
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    def __init__(self, dominant_weight):
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        """
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        EXAMPLES::
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            sage: C=CartanType(['E',6])
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            sage: La=C.root_system().weight_lattice().fundamental_weights()
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            sage: p2=2*La[2]
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            sage: p1=La[2]
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            sage: p0=0*La[2]
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            sage: T = HighestWeightCrystal(0*La[2])
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            sage: T.cardinality()
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            1
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            sage: T = HighestWeightCrystal(La[2])
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            sage: T.cardinality()
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            78
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            sage: T = HighestWeightCrystal(2*La[2])
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            sage: T.cardinality()
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            2430
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        """
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        B1 = CrystalOfLetters(['E',6])
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        B6 = CrystalOfLetters(['E',6], dual = True)
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        self.column_crystal = {1 : B1, 6 : B6,
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                               4 : TensorProductOfCrystals(B1,B1,B1,generators=[[B1([-3,4]),B1([-1,3]),B1([1])]]),
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                               3 : TensorProductOfCrystals(B1,B1,generators=[[B1([-1,3]),B1([1])]]),
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                               5 : TensorProductOfCrystals(B6,B6,generators=[[B6([5,-6]),B6([6])]]),
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                               2 : TensorProductOfCrystals(B6,B1,generators=[[B6([2,-1]),B1([1])]])}
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        FiniteDimensionalHighestWeightCrystal_TypeE.__init__(self, dominant_weight)
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class FiniteDimensionalHighestWeightCrystal_TypeE7(FiniteDimensionalHighestWeightCrystal_TypeE):
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    r"""
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    Class of finite dimensional highest weight crystals of type `E_7`.
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    EXAMPLES::
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        sage: C=CartanType(['E',7])
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        sage: La=C.root_system().weight_lattice().fundamental_weights()
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        sage: T = HighestWeightCrystal(La[1])
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        sage: T.cardinality()
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        133
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        sage: B7 = T.column_crystal[7]; B7
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        The crystal of letters for type ['E', 7]
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        sage: t = T(B7([-5, 6]), B7([-2, 3])); t
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        [[-5, 6], [-2, 3]]
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        sage: [t.epsilon(i) for i in T.index_set()]
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        [0, 1, 0, 0, 1, 0, 0]
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        sage: [t.phi(i) for i in T.index_set()]
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        [0, 0, 1, 0, 0, 1, 0]
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        sage: TestSuite(t).run()
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    """
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    def __init__(self, dominant_weight):
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        """
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        EXAMPLES::
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            sage: C=CartanType(['E',7])
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            sage: La=C.root_system().weight_lattice().fundamental_weights()
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            sage: T = HighestWeightCrystal(0*La[1])
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            sage: T.cardinality()
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            1
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            sage: T = HighestWeightCrystal(La[1])
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            sage: T.cardinality()
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            133
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            sage: T = HighestWeightCrystal(2*La[1])
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            sage: T.cardinality()
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            7371
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        """
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        B = CrystalOfLetters(['E',7])
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        self.column_crystal = {7 : B,
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                               1 : TensorProductOfCrystals(B,B,generators=[[B([-7,1]),B([7])]]),
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                               2 : TensorProductOfCrystals(B,B,B,generators=[[B([-1,2]),B([-7,1]),B([7])]]),
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                               3 : TensorProductOfCrystals(B,B,B,B,generators=[[B([-2,3]),B([-1,2]),B([-7,1]),B([7])]]),
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                               4 : TensorProductOfCrystals(B,B,B,B,generators=[[B([-5,4]),B([-6,5]),B([-7,6]),B([7])]]),
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                               5 : TensorProductOfCrystals(B,B,B,generators=[[B([-6,5]),B([-7,6]),B([7])]]),
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                               6 : TensorProductOfCrystals(B,B,generators=[[B([-7,6]),B([7])]])}
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        FiniteDimensionalHighestWeightCrystal_TypeE.__init__(self, dominant_weight)
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