Python RootSystem.index_set方法代码示例

# 需要导入模块: from sage.combinat.root_system.root_system import RootSystem [as 别名]
# 或者: from sage.combinat.root_system.root_system.RootSystem import index_set [as 别名]
    def __classcall_private__(cls, starting_weight, cartan_type = None, starting_weight_parent = None):
        """
        Classcall to mend the input.

        Internally, the
        :class:`~sage.combinat.crystals.littlemann_path.CrystalOfLSPaths` code
        works with a ``starting_weight`` that is in the weight space associated
        to the crystal. The user can, however, also input a ``cartan_type``
        and the coefficients of the fundamental weights as
        ``starting_weight``. This code transforms the input into the right
        format (also necessary for UniqueRepresentation).

        TESTS::

            sage: crystals.LSPaths(['A',2,1],[-1,0,1])
            The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]

            sage: R = RootSystem(['B',2,1])
            sage: La = R.weight_space(extended=True).basis()
            sage: C = crystals.LSPaths(['B',2,1],[0,0,1])
            sage: B = crystals.LSPaths(La[2])
            sage: B is C
            True
        """
        if cartan_type is not None:
            cartan_type, starting_weight = CartanType(starting_weight), cartan_type
            if cartan_type.is_affine():
                extended = True
            else:
                extended = False

            R = RootSystem(cartan_type)
            P = R.weight_space(extended = extended)
            Lambda = P.basis()
            offset = R.index_set()[Integer(0)]
            starting_weight = P.sum(starting_weight[j-offset]*Lambda[j] for j in R.index_set())
        if starting_weight_parent is None:
            starting_weight_parent = starting_weight.parent()
        else:
            # Both the weight and the parent of the weight are passed as arguments of init to be able
            # to distinguish between crystals with the extended and non-extended weight lattice!
            if starting_weight.parent() != starting_weight_parent:
                raise ValueError("The passed parent is not equal to parent of the inputted weight!")

        return super(CrystalOfLSPaths, cls).__classcall__(cls, starting_weight, starting_weight_parent = starting_weight_parent)

# 需要导入模块: from sage.combinat.root_system.root_system import RootSystem [as 别名]
# 或者: from sage.combinat.root_system.root_system.RootSystem import index_set [as 别名]
    def __classcall_private__(cls, starting_weight, cartan_type = None):
        """
        Classcall to mend the input.

        Internally, the CrystalOfLSPaths code works with a ``starting_weight`` that
        is in the ``weight_space`` associated to the crystal. The user can, however,
        also input a ``cartan_type`` and the coefficients of the fundamental weights
        as ``starting_weight``. This code transforms the input into the right
        format (also necessary for UniqueRepresentation).

        TESTS::

            sage: CrystalOfLSPaths(['A',2,1],[-1,0,1])
            The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[2]

            sage: R = RootSystem(['B',2,1])
            sage: La = R.weight_space().basis()
            sage: C = CrystalOfLSPaths(['B',2,1],[0,0,1])
            sage: B = CrystalOfLSPaths(La[2])
            sage: B is C
            True
        """
        if cartan_type is not None:
            cartan_type, starting_weight = CartanType(starting_weight), cartan_type
            if cartan_type.is_affine():
                extended = True
            else:
                extended = False

            R = RootSystem(cartan_type)
            P = R.weight_space(extended = extended)
            Lambda = P.basis()
            offset = R.index_set()[Integer(0)]
            starting_weight = P.sum(starting_weight[j-offset]*Lambda[j] for j in R.index_set())

        return super(CrystalOfLSPaths, cls).__classcall__(cls, starting_weight)

# 需要导入模块: from sage.combinat.root_system.root_system import RootSystem [as 别名]
# 或者: from sage.combinat.root_system.root_system.RootSystem import index_set [as 别名]
class CrystalOfLSPaths(UniqueRepresentation, Parent):
    r"""
    Crystal graph of LS paths generated from the straight-line path to a given weight.

    INPUT:

    - ``cartan_type`` -- the Cartan type of a finite or affine root system
    - ``starting_weight`` -- a weight given as a list of coefficients of the fundamental weights

    The crystal class of piecewise linear paths in the weight space,
    generated from a straight-line path from the origin to a given
    element of the weight lattice.

    OUTPUT: - a tuple of weights defining the directions of the piecewise linear segments

    EXAMPLES::

        sage: C = CrystalOfLSPaths(['A',2,1],[-1,0,1]); C
        The crystal of LS paths of type ['A', 2, 1] and weight (-1, 0, 1)
        sage: c = C.module_generators[0]; c
        (-Lambda[0] + Lambda[2],)
        sage: [c.f(i) for i in C.index_set()]
        [None, None, (Lambda[1] - Lambda[2],)]

        sage: R = C.R; R
        Root system of type ['A', 2, 1]
        sage: Lambda = R.weight_space().basis(); Lambda
        Finite family {0: Lambda[0], 1: Lambda[1], 2: Lambda[2]}
        sage: b=C(tuple([-Lambda[0]+Lambda[2]]))
        sage: b==c
        True
        sage: b.f(2)
        (Lambda[1] - Lambda[2],)

    For classical highest weight crystals we can also compare the results with the tableaux implementation::

        sage: C = CrystalOfLSPaths(['A',2],[1,1])
        sage: list(set(C.list()))
        [(-Lambda[1] - Lambda[2],), (-Lambda[1] + 1/2*Lambda[2], Lambda[1] - 1/2*Lambda[2]), (-Lambda[1] + 2*Lambda[2],),
        (1/2*Lambda[1] - Lambda[2], -1/2*Lambda[1] + Lambda[2]), (Lambda[1] - 2*Lambda[2],), (-2*Lambda[1] + Lambda[2],),
        (2*Lambda[1] - Lambda[2],), (Lambda[1] + Lambda[2],)]
        sage: C.cardinality()
        8
        sage: B = CrystalOfTableaux(['A',2],shape=[2,1])
        sage: B.cardinality()
        8
        sage: B.digraph().is_isomorphic(C.digraph())
        True

    TESTS::

        sage: C = CrystalOfLSPaths(['A',2,1],[-1,0,1])
        sage: TestSuite(C).run(skip=['_test_elements', '_test_elements_eq', '_test_enumerated_set_contains', '_test_some_elements'])
        sage: C = CrystalOfLSPaths(['E',6],[1,0,0,0,0,0])
        sage: TestSuite(C).run()

    REFERENCES::

        .. [L] P. Littelmann, Paths and root operators in representation theory. Ann. of Math. (2) 142 (1995), no. 3, 499-525.
    """

    @staticmethod
    def __classcall__(cls, cartan_type, starting_weight):
        """
        cartan_type and starting_weight are lists, which are mutable. The class
        UniqueRepresentation requires immutable inputs. The following code
        fixes this problem.

        TESTS::

            sage: CrystalOfLSPaths.__classcall__(CrystalOfLSPaths,['A',2,1],[-1,0,1])
            The crystal of LS paths of type ['A', 2, 1] and weight (-1, 0, 1)
        """
        cartan_type = CartanType(cartan_type)
        starting_weight = tuple(starting_weight)
        return super(CrystalOfLSPaths, cls).__classcall__(cls, cartan_type, starting_weight)

    def __init__(self, cartan_type, starting_weight):
        """
        EXAMPLES::

            sage: C = CrystalOfLSPaths(['A',2,1],[-1,0,1]); C
            The crystal of LS paths of type ['A', 2, 1] and weight (-1, 0, 1)
            sage: C.R
            Root system of type ['A', 2, 1]
            sage: C.weight
            -Lambda[0] + Lambda[2]
            sage: C.weight.parent()
            Extended weight space over the Rational Field of the Root system of type ['A', 2, 1]
            sage: C.module_generators
            [(-Lambda[0] + Lambda[2],)]
        """
        self._cartan_type = CartanType(cartan_type)
        self.R = RootSystem(cartan_type)

        self._name = "The crystal of LS paths of type %s and weight %s"%(cartan_type,starting_weight)

        if self._cartan_type.is_affine():
            self.extended = True
            if all(i>=0 for i in starting_weight):
#.........这里部分代码省略.........

# 需要导入模块: from sage.combinat.root_system.root_system import RootSystem [as 别名]
# 或者: from sage.combinat.root_system.root_system.RootSystem import index_set [as 别名]

#.........这里部分代码省略.........
        causes a problem for class UniqueRepresentation, the following code
        fixes this problem.

        EXAMPLES::
            sage: ClassicalCrystalOfAlcovePaths.__classcall__(ClassicalCrystalOfAlcovePaths,['A',3],[0,1,0])
            <class 'sage.combinat.crystals.alcove_path.ClassicalCrystalOfAlcovePaths_with_category'>
        """
        cartan_type = CartanType(cartan_type)
        highest_weight = tuple(highest_weight)
        return super(ClassicalCrystalOfAlcovePaths, cls).__classcall__(cls, cartan_type, highest_weight)

    def __init__(self, cartan_type, highest_weight):
        """
        EXAMPLES::

            sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[1,0,0])
            sage: C.list()
            [[], [0], [0, 1], [0, 1, 2]]
            sage: TestSuite(C).run()
        """
        Parent.__init__(self, category = ClassicalCrystals())
        self._cartan_type = CartanType(cartan_type)
        self._name = "The crystal of alcove paths for type %s"%cartan_type
        self.chain_cache = {}
        self.endweight_cache = {}

        self.R = RootSystem(cartan_type)
        alpha = self.R.root_space().simple_roots()
        Lambda = self.R.weight_space().basis()

        self.positive_roots = sorted(self.R.root_space().positive_roots());

        self.weight = Lambda[Integer(1)] - Lambda[Integer(1)]
        offset = self.R.index_set()[Integer(0)]
        for j in self.R.index_set():
            self.weight = self.weight + highest_weight[j-offset]*Lambda[j]

        self.initial_element = self([])

        self.initial_element.chain = self.get_initial_chain(self.weight)
        rho = (Integer(1)/Integer(2))*sum(self.positive_roots)
        self.initial_element.endweight = rho

        self.chain_cache[ str([]) ] = self.initial_element.chain
        self.endweight_cache[ str([]) ] = self.initial_element.endweight

        self.module_generators = [self.initial_element]

        self._list = super(ClassicalCrystalOfAlcovePaths, self).list()
        self._digraph = super(ClassicalCrystalOfAlcovePaths, self).digraph()
        self._digraph_closure = self.digraph().transitive_closure()

    def get_initial_chain(self, highest_weight):
        """
        Called internally by __init__() to construct the chain of roots
        associated to the highest weight element.

        EXAMPLES::
            sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
            sage: C.get_initial_chain(RootSystem(['A',3]).weight_space().basis()[1])
            [[alpha[1], alpha[1]], [alpha[1] + alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
        """
        pos_roots = self.positive_roots
        tis = self.R.index_set()
        tis.reverse()
        cv_to_pos_root = {}