"""
Root system data for affine Cartan types
"""
#*****************************************************************************
#       Copyright (C) 2013 Nicolas M. Thiery <nthiery at users.sf.net>,
#
#  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.misc.lazy_attribute import lazy_attribute
from sage.combinat.free_module import CombinatorialFreeModule
from weight_lattice_realizations import WeightLatticeRealizations

class AmbientSpace(CombinatorialFreeModule):
    r"""
    Ambient space for affine types.

    This is constructed from the data in the corresponding classical
    ambient space. Namely, this space is obtained by adding two
    elements `\delta` and `\delta^\vee` to the basis of the classical
    ambient space, and by endowing it with the canonical scalar product.

    The coefficient of an element in `\delta^\vee`, thus its scalar
    product with `\delta^\vee` gives its level, and dualy for the
    colevel. The canonical projection onto the classical ambient space
    (by killing `\delta` and `\delta^\vee`) maps the simple roots
    (except `\alpha_0`) onto the corresponding classical simple roots,
    and similarly for the coroots, fundamental weights, ...
    Altogether, this uniquely determines the embedding of the root,
    coroot, weight, and coweight lattices. See :meth:`simple_root` and
    :meth:`fundamental_weight` for the details.

    .. WARNING::

        In type `BC`, the null root is in fact::

            sage: R = RootSystem(["BC",3,2]).ambient_space()
            sage: R.null_root()
            2*e['delta']

    .. WARNING::

        In the literature one often considers a larger affine ambient
        space obtained from the classical ambient space by adding four
        dimensions, namely for the fundamental weight `\Lambda_0` the
        fundamental coweight `\Lambda^\vee_0`, the null root `\delta`,
        and the null coroot `c` (aka central element).  In this larger
        ambient space, the scalar product is degenerate: `\langle
        \delta,\delta\rangle=0` and similarly for the null coroot.

        In the current implementation, `\Lambda_0` and the null coroot
        are identified:

            sage: L = RootSystem(["A",3,1]).ambient_space()
            sage: Lambda = L.fundamental_weights()
            sage: Lambda[0]
            e['deltacheck']
            sage: L.null_coroot()
            e['deltacheck']

        Therefore the scalar product of the null coroot with itself
        differs from the larger ambient space::

            sage: L.null_coroot().scalar(L.null_coroot())
            1

        In general, scalar products between two elements that do not
        live on "opposite sides" won't necessarily match.

    EXAMPLES::

        sage: R = RootSystem(["A",3,1])
        sage: e = R.ambient_space(); e
        Ambient space of the Root system of type ['A', 3, 1]
        sage: TestSuite(e).run()

    Systematic checks on all affine types::

        sage: for ct in CartanType.samples(affine=True, crystallographic=True):
        ...       if ct.classical().root_system().ambient_space() is not None:
        ...           print ct
        ...           L = ct.root_system().ambient_space()
        ...           assert L
        ...           TestSuite(L).run()
        ['A', 1, 1]
        ['A', 5, 1]
        ['B', 1, 1]
        ['B', 5, 1]
        ['C', 1, 1]
        ['C', 5, 1]
        ['D', 3, 1]
        ['D', 5, 1]
        ['E', 6, 1]
        ['E', 7, 1]
        ['E', 8, 1]
        ['F', 4, 1]
        ['G', 2, 1]
        ['BC', 1, 2]
        ['BC', 5, 2]
        ['B', 5, 1]^*
        ['C', 4, 1]^*
        ['F', 4, 1]^*
        ['G', 2, 1]^*
        ['BC', 1, 2]^*
        ['BC', 5, 2]^*
    """
    @classmethod
    def smallest_base_ring(cls, cartan_type):
        r"""
        Return the smallest base ring the ambient space can be defined on.

        This is the smallest base ring for the associated classical
        ambient space.

        .. SEEALSO:: :meth:`~sage.combinat.root_system.ambient_space.AmbientSpace.smallest_base_ring`

        EXAMPLES::

            sage: cartan_type = CartanType(["A",3,1])
            sage: cartan_type.AmbientSpace.smallest_base_ring(cartan_type)
            Integer Ring
            sage: cartan_type = CartanType(["B",3,1])
            sage: cartan_type.AmbientSpace.smallest_base_ring(cartan_type)
            Rational Field
        """
        classical = cartan_type.classical()
        return cartan_type.classical().root_system().ambient_space().smallest_base_ring(classical)

    def __init__(self, root_system, base_ring):
        r"""
        EXAMPLES::

            sage: R = RootSystem(["A",3,1])
            sage: R.cartan_type().AmbientSpace
            <class 'sage.combinat.root_system.type_affine.AmbientSpace'>
            sage: e = R.ambient_space(); e
            Ambient space of the Root system of type ['A', 3, 1]
            sage: TestSuite(R.ambient_space()).run()

            sage: L = RootSystem(['A',3]).coroot_lattice()
            sage: e.has_coerce_map_from(L)
            True
            sage: e(L.simple_root(1))
            e[0] - e[1]
        """
        self.root_system = root_system
        classical = root_system.cartan_type().classical().root_system().ambient_space(base_ring)
        basis_keys = tuple(classical.basis().keys()) + ("delta","deltacheck")
        CombinatorialFreeModule.__init__(self, base_ring,
                                         basis_keys,
                                         prefix = "e",
                                         latex_prefix = "e",
                                         category = WeightLatticeRealizations(base_ring))
        self._weight_space = self.root_system.weight_space(base_ring=base_ring,extended=True)
        self.classical().module_morphism(self.monomial, codomain=self).register_as_coercion()
        # Duplicated from ambient_space.AmbientSpace
        coroot_lattice = self.root_system.coroot_lattice()
        coroot_lattice.module_morphism(self.simple_coroot, codomain=self).register_as_coercion()

    def _name_string(self, capitalize=True, base_ring=False, type=True):
        r"""
        Utility to implement _repr_

        EXAMPLES::

            sage: RootSystem(['A',4,1]).ambient_lattice()
            Ambient lattice of the Root system of type ['A', 4, 1]
            sage: RootSystem(['A',4,1]).ambient_space()
            Ambient space of the Root system of type ['A', 4, 1]
            sage: RootSystem(['A',4,1]).dual.ambient_lattice()
            Coambient lattice of the Root system of type ['A', 4, 1]

            sage: RootSystem(['A',4,1]).ambient_lattice()._repr_()
            "Ambient lattice of the Root system of type ['A', 4, 1]"
            sage: RootSystem(['A',4,1]).ambient_lattice()._name_string()
            "Ambient lattice of the Root system of type ['A', 4, 1]"
        """
        return self._name_string_helper("ambient", capitalize=capitalize, base_ring=base_ring, type=type)

    _repr_ = _name_string

    @cached_method
    def _to_classical_on_basis(self, i):
        r"""
        Implement the projection onto the corresponding classical space or lattice, on the basis.

        INPUT:

        - ``i`` -- the index of an element of the basis of ``self``,
          namely 0, 1, 2, ..., "delta", or "deltacheck"

        EXAMPLES::

            sage: L = RootSystem(["A",2,1]).ambient_space()
            sage: L._to_classical_on_basis("delta")
            (0, 0, 0)
            sage: L._to_classical_on_basis("deltacheck")
            (0, 0, 0)
            sage: L._to_classical_on_basis(0)
            (1, 0, 0)
            sage: L._to_classical_on_basis(1)
            (0, 1, 0)
            sage: L._to_classical_on_basis(2)
            (0, 0, 1)
        """
        if i=="delta" or i=="deltacheck":
            return self.classical().zero()
        else:
            return self.classical().monomial(i)

    def is_extended(self):
        r"""
        Return whether this is a realization of the extended weight lattice: yes!

        .. SEEALSO::

            - :class:`sage.combinat.root_system.weight_space.WeightSpace`
            - :meth:`sage.combinat.root_sytem.weight_lattice_realizations.WeightLatticeRealizations.ParentMethods.is_extended`

        EXAMPLES::

            sage: RootSystem(['A',3,1]).ambient_space().is_extended()
            True
        """
        return True

    @cached_method
    def fundamental_weight(self, i):
        r"""
        Return the fundamental weight `\Lambda_i` in this ambient space.

        It is constructed by taking the corresponding fundamental
        weight of the classical ambient space (or `0` for `\Lambda_0`)
        and raising it to the appropriate level by adding a suitable
        multiple of `\delta^\vee`.

        EXAMPLES::

            sage: RootSystem(['A',3,1]).ambient_space().fundamental_weight(2)
            e[0] + e[1] + e['deltacheck']
            sage: RootSystem(['A',3,1]).ambient_space().fundamental_weights()
            Finite family {0: e['deltacheck'], 1: e[0] + e['deltacheck'],
                           2: e[0] + e[1] + e['deltacheck'], 3: e[0] + e[1] + e[2] + e['deltacheck']}
            sage: RootSystem(['A',3]).ambient_space().fundamental_weights()
            Finite family {1: (1, 0, 0, 0), 2: (1, 1, 0, 0), 3: (1, 1, 1, 0)}
            sage: RootSystem(['A',3,1]).weight_lattice().fundamental_weights().map(attrcall("level"))
            Finite family {0: 1, 1: 1, 2: 1, 3: 1}

            sage: RootSystem(['B',3,1]).ambient_space().fundamental_weights()
            Finite family {0: e['deltacheck'], 1: e[0] + e['deltacheck'],
                           2: e[0] + e[1] + 2*e['deltacheck'], 3: 1/2*e[0] + 1/2*e[1] + 1/2*e[2] + e['deltacheck']}
            sage: RootSystem(['B',3]).ambient_space().fundamental_weights()
            Finite family {1: (1, 0, 0), 2: (1, 1, 0), 3: (1/2, 1/2, 1/2)}
            sage: RootSystem(['B',3,1]).weight_lattice().fundamental_weights().map(attrcall("level"))
            Finite family {0: 1, 1: 1, 2: 2, 3: 1}

       In type `BC` dual, the coefficient of '\delta^\vee' is the level
       divided by `2` to take into account that the null coroot is
       `2\delta^\vee`::

            sage: R = CartanType(['BC',3,2]).dual().root_system()
            sage: R.ambient_space().fundamental_weights()
            Finite family {0: e['deltacheck'], 1: e[0] + e['deltacheck'],
                           2: e[0] + e[1] + e['deltacheck'],
                           3: 1/2*e[0] + 1/2*e[1] + 1/2*e[2] + 1/2*e['deltacheck']}
            sage: R.weight_lattice().fundamental_weights().map(attrcall("level"))
            Finite family {0: 2, 1: 2, 2: 2, 3: 1}
            sage: R.ambient_space().null_coroot()
            2*e['deltacheck']

        By a slight naming abuse this function also accepts "delta" as
        input so that it can be used to implement the embedding from
        the extended weight lattice::

            sage: RootSystem(['A',3,1]).ambient_space().fundamental_weight("delta")
            e['delta']
        """
        if i == "delta":
            return self.monomial("delta")
        deltacheck = self.monomial("deltacheck")
        result = deltacheck * self._weight_space.fundamental_weight(i).level() / deltacheck.level()
        if i != self.cartan_type().special_node():
            result += self(self.classical().fundamental_weight(i))
        return result

    @cached_method
    def simple_root(self, i):
        r"""
        Return the `i`-th simple root of this affine ambient space.

        EXAMPLES:

        It is built straightforwardly from the corresponding simple
        root `\alpha_i` in the classical ambient space::

            sage: RootSystem(["A",3,1]).ambient_space().simple_root(1)
            e[0] - e[1]

        For the special node (typically `i=0`), `\alpha_0` is built
        from the other simple roots using the column annihilator of
        the Cartan matrix and adding `\delta`, where `\delta` is the
        null root::

            sage: RootSystem(["A",3]).ambient_space().simple_roots()
            Finite family {1: (1, -1, 0, 0), 2: (0, 1, -1, 0), 3: (0, 0, 1, -1)}
            sage: RootSystem(["A",3,1]).ambient_space().simple_roots()
            Finite family {0: -e[0] + e[3] + e['delta'], 1: e[0] - e[1], 2: e[1] - e[2], 3: e[2] - e[3]}

        Here is a twisted affine example::

            sage: RootSystem(CartanType(["B",3,1]).dual()).ambient_space().simple_roots()
            Finite family {0: -e[0] - e[1] + e['delta'], 1: e[0] - e[1], 2: e[1] - e[2], 3: 2*e[2]}

        In fact `\delta` is really `1/a_0` times the null root (see
        the discussion in :class:`~sage.combinat.root_system.weight_space.WeightSpace`)
        but this only makes a difference in type `BC`::

            sage: L = RootSystem(CartanType(["BC",3,2])).ambient_space()
            sage: L.simple_roots()
            Finite family {0: -e[0] + e['delta'], 1: e[0] - e[1], 2: e[1] - e[2], 3: 2*e[2]}
            sage: L.null_root()
            2*e['delta']

        .. NOTE::

            An alternative would have been to use the default
            implementation of the simple roots as linear combinations
            of the fundamental weights. However, as in type `A_n` it is
            preferable to take a slight variant to avoid rational
            coefficient (the usual `GL_n` vs `SL_n` issue).

        .. SEEALSO::

            - :meth:`~sage.combinat.root_system.weight_space.WeightSpace.simple_root`
            - :class:`~sage.combinat.root_system.weight_space.WeightSpace`
            - :meth:`CartanType.col_annihilator`
            - :meth:`null_root`
        """
        cartan_type  = self.cartan_type()
        special_node = cartan_type.special_node()
        if i == special_node:
            return self(self._classical_alpha_0()) + self.monomial("delta")
        else:
            return self(self.classical().simple_root(i))

    @cached_method
    def simple_coroot(self, i):
        r"""
        Return the `i`-th simple coroot `\alpha_i^\vee` of this affine ambient space.

        EXAMPLES::

            sage: RootSystem(["A",3,1]).ambient_space().simple_coroot(1)
            e[0] - e[1]

        It is built as the coroot associated to the simple root
        `\alpha_i`::

            sage: RootSystem(["B",3,1]).ambient_space().simple_roots()
            Finite family {0: -e[0] - e[1] + e['delta'], 1: e[0] - e[1], 2: e[1] - e[2], 3: e[2]}
            sage: RootSystem(["B",3,1]).ambient_space().simple_coroots()
            Finite family {0: -e[0] - e[1] + e['deltacheck'], 1: e[0] - e[1], 2: e[1] - e[2], 3: 2*e[2]}

        .. TODO:: Factor out this code with the classical ambient space.
        """
        return self.simple_root(i).associated_coroot()

    def coroot_lattice(self):
        """
        EXAMPLES::

            sage: RootSystem(["A",3,1]).ambient_lattice().coroot_lattice()
            Ambient lattice of the Root system of type ['A', 3, 1]

        .. TODO:: Factor out this code with the classical ambient space.
        """
        return self

    def _plot_projection(self, x):
        """
        Implements the default projection to be used for plots

        For affine ambient spaces, the default implementation is to
        project onto the classical coordinates according to the
        default projection for the classical ambient space, while
        keeping an extra coordinate for the coefficient of
        `\delta^\vee` to keep the level information.

        .. SEEALSO:: :meth:`sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations._plot_projection`

        EXAMPLES::

            sage: L = RootSystem(["B",2,1]).ambient_space()
            sage: e = L.basis()
            sage: L._plot_projection(e[0])
            (1, 0, 0)
            sage: L._plot_projection(e[1])
            (0, 1, 0)
            sage: L._plot_projection(e["delta"])
            (0, 0, 0)
            sage: L._plot_projection(e["deltacheck"])
            (0, 0, 1)

            sage: L = RootSystem(["A",2,1]).ambient_space()
            sage: e = L.basis()
            sage: L._plot_projection(e[0])
            (1/2, 989/1142, 0)
            sage: L._plot_projection(e[1])
            (-1, 0, 0)
            sage: L._plot_projection(e["delta"])
            (0, 0, 0)
            sage: L._plot_projection(e["deltacheck"])
            (0, 0, 1)
        """
        from sage.modules.free_module_element import vector
        classical = self.classical()
        # Any better way to concatenate two vectors?
        return vector(list(vector(classical._plot_projection(classical(x)))) +
                      [x["deltacheck"]])

    class Element(CombinatorialFreeModule.Element):

        def inner_product(self, other):
            r"""
            Implement the canonical inner product of ``self`` with ``other``.

            EXAMPLES::

                sage: e = RootSystem(['B',3,1]).ambient_space()
                sage: B = e.basis()
                sage: matrix([[x.inner_product(y) for x in B] for y in B])
                [1 0 0 0 0]
                [0 1 0 0 0]
                [0 0 1 0 0]
                [0 0 0 1 0]
                [0 0 0 0 1]
                sage: x = e.an_element(); x
                2*e[0] + 2*e[1] + 3*e[2]
                sage: x.inner_product(x)
                17

            :meth:`scalar` is an alias for this method::

                sage: x.scalar(x)
                17

            .. TODO:: Lift to CombinatorialFreeModule.Element as canonical_inner_product
            """
            if self.parent() is not other.parent():
                raise TypeError("the parents must be the same")
            return self.base_ring().sum( self[i] * c for (i,c) in other )

        scalar = inner_product

        def associated_coroot(self):
            r"""
            Return the coroot associated to ``self``.

            INPUT:

            - ``self`` -- a root

            EXAMPLES::

                sage: alpha = RootSystem(['C',2,1]).ambient_space().simple_roots()
                sage: alpha
                Finite family {0: -2*e[0] + e['delta'], 1: e[0] - e[1], 2: 2*e[1]}
                sage: alpha[0].associated_coroot()
                -e[0] + e['deltacheck']
                sage: alpha[1].associated_coroot()
                e[0] - e[1]
                sage: alpha[2].associated_coroot()
                e[1]
            """
            # CHECKME: does it make any sense to not rescale the delta term?
            L = self.parent()
            c = self["delta"]
            self = self - L.term("delta", c)
            return (2*self) / self.inner_product(self) + L.term("deltacheck", c)
