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r""" Virtual Crystals
These are the crystals that are subsets of a larger ambient crystal with virtual crystal operators.
AUTHORS:
- Travis Scrimshaw (2013-10-16): Initial implementation """
#***************************************************************************** # Copyright (C) 2013 Travis Scrimshaw <tscrim at ucdavis.edu> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #****************************************************************************
r""" A virtual crystal `V` of an ambient crystal `\widehat{B}` is a crystal formed by taking a subset of `\widehat{B}` and whose crystal structure is given by
.. MATH::
e_i = \prod_{j \in \sigma_i} \widehat{e}_j^{\gamma_i}, \quad f_i = \prod_{j \in \sigma_i} \widehat{f}_j^{\gamma_i},
.. MATH::
\varepsilon_i = \frac{\widehat{\varepsilon}_j}{\gamma_j}, \quad \varphi_i = \frac{\widehat{\varphi}_j}{\gamma_j}, \quad \operatorname{wt} = \Psi^{-1} \circ \widehat{\operatorname{wt}}
where `\sigma_i` is a subset of the index set of `B`, `\gamma_i \in \ZZ` are the *scaling factors*, and `\Psi : P \to \widehat{P}` is an embedding of the weight lattices. We note that for the crystal to be well-defined, we must have
.. MATH::
\widehat{\varepsilon}_j = \widehat{\varepsilon|j^{\prime}}, \quad \widehat{\varphi}_j = \widehat{\varphi}_{j^{\prime}}
for all `j, j^{\prime} \in \sigma_i` and that the order that the Kashiwara operators in the ambient space are applied does not affect the result.
INPUT:
- ``ambient`` -- the ambient crystal - ``virtualization`` -- a dictionary whose key `i` corresponds to the set `\sigma_i` - ``scaling_factors`` -- a dictionary whose key `i` corresponds to the scaling factor `\gamma_i` - ``contained`` -- (optional) a set (or function) which specifies when an element is contained in the subcrystal; the default is everything possible is included - ``generators`` -- (optional) the generators for the virtual crystal; the default is the generators for the ambient crystal - ``cartan_type`` -- (optional) the Cartan type for the virtual crystal; the default is the Cartan type for the ambient crystal - ``index_set`` -- (optional) the index set for the virtual crystal; the default is the index set for the Cartan type - ``category`` -- (optional) the category for the virtual crystal; the default is the :class:`~sage.categories.crystals.Crystals` category
EXAMPLES:
We construct an example from a natural virtualization map of type `C_n` in type `A_{2n-1}`::
sage: C = crystals.Tableaux(['C',2], shape=[1]) sage: A = crystals.Tableaux(['A',3], shape=[2,1,1]) sage: psi = C.crystal_morphism(A.module_generators) sage: V = psi.image() sage: list(V) [[[1, 1], [2], [3]], [[1, 2], [2], [4]], [[1, 3], [3], [4]], [[2, 4], [3], [4]]] sage: V.digraph().is_isomorphic(C.digraph(), edge_labels=True) True
We construct the virtualization of a `U_q'(\mathfrak{g})`-crystal `B^{r,s}` of type `C_n^{(1)}` in type `A_{2n+1}^{(2)}`. Here it is not a default folding known to Sage, so we have to explicitly state the folding (since the scaling factors are not specified, they are all assumed to be 1)::
sage: K = crystals.KirillovReshetikhin(['C',2,1], 1,1) sage: VK = crystals.KirillovReshetikhin(['A',5,2], 1,1) sage: target = VK.module_generator().f(1); target [[2]] sage: psi = K.crystal_morphism({K.module_generator(): target}, ....: virtualization={0:[0,1], 1:[2], 2:[3]}) sage: V = psi.image() sage: list(V) [[[2]], [[3]], [[-2]], [[-3]]] sage: V.digraph().is_isomorphic(K.digraph(), edge_labels=True) True
We create an example of `B(\Lambda_n)` of type `B_n` inside of `B(2\Lambda_n)` using the doubling map through the (virtual) subcrystal method::
sage: BB = crystals.Tableaux(['B',3], shape=[1,1,1]) sage: S = BB.subcrystal(scaling_factors={1:2, 2:2, 3:2}) sage: B = crystals.Tableaux(['B',3], shape=[1/2,1/2,1/2]) sage: S.digraph().is_isomorphic(B.digraph(), edge_labels=True) True
We can also directly construct a virtual crystal using :class:`VirtualCrystal` (however it is recommended to use either :meth:`~sage.categories.crystals.Crystals.ParentMethods.crystal_morphism` or :meth:`~sage.categories.crystals.Crystals.ParentMethods.subcrystal`)::
sage: from sage.combinat.crystals.virtual_crystal import VirtualCrystal sage: A = crystals.Tableaux(['A',3], shape=[2,1,1]) sage: V = VirtualCrystal(A, {1:(1,3), 2:(2,)}, {1:1, 2:2}, cartan_type=['C',2]) sage: G = crystals.Tableaux(['C',2], shape=[1]).digraph() sage: V.digraph().is_isomorphic(G, edge_labels=True) True
sage: C1 = crystals.Tableaux(['A',3], shape=[1]) sage: C2 = crystals.Tableaux(['A',3], shape=[1,1,1]) sage: T = C1.tensor(C2) sage: mg = T(C1.module_generators[0], C2.module_generators[0]) sage: V = VirtualCrystal(A, {1:(1,3), 2:(2,)}, {1:1, 2:2}, ....: cartan_type=['C',2], generators=[mg]) sage: V.digraph().is_isomorphic(G, edge_labels=True) True
REFERENCES:
- [FOS09]_ - [OSS03]_ - [OSS2003]_ """ contained=None, generators=None, cartan_type=None, index_set=None, category=None): """ Normalize arguments to ensure a unique representation.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi1 = B.crystal_morphism(C.module_generators) sage: V1 = psi1.image() sage: psi2 = B.crystal_morphism(C.module_generators, index_set=[1,2,3]) sage: V2 = psi2.image() sage: V1 is V2 True
TESTS:
Check that :trac:`19481` is fixed::
sage: from sage.combinat.crystals.virtual_crystal import VirtualCrystal sage: A = crystals.Tableaux(['A',3], shape=[2,1,1]) sage: V = VirtualCrystal(A, {1:(1,3), 2:(2,)}, {1:1, 2:2}, cartan_type=['C',2]) sage: V.category() Category of finite crystals """ cartan_type = ambient.cartan_type() else:
contained, tuple(generators), cartan_type, tuple(index_set), category)
contained, generators, cartan_type, index_set, category): """ Initialize ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: TestSuite(V).run() """ cartan_type, index_set, category)
""" Return a string representation of ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: psi.image() Virtual crystal of The crystal of tableaux of type ['D', 4] and shape(s) [[2]] of type ['B', 3] """
""" Check if ``x`` is in ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: mg = C.module_generators[0] sage: mg in V True sage: mg.f(1) in V False sage: mg.f(1).f(1) in V True """ x = self.element_class(self, self._ambient(x)) x = self.element_class(self, x.value) return True
""" Return the virtualization sets `\sigma_i`.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: V.virtualization() Finite family {1: (1,), 2: (2,), 3: (3, 4)} """
""" Return the scaling factors `\gamma_i`.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: V.scaling_factors() Finite family {1: 2, 2: 2, 3: 1} """
""" An element of a virtual (sub)crystal. Wraps an element in the ambient crystal. """ """ Return `e_i` of ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: mg = V.module_generators[0] sage: mg.e(1) sage: b = psi(B.module_generators[0].f(1)) sage: V(b).e(1) [[1, 1]] """
""" Return `f_i` of ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: mg = V.module_generators[0] sage: mg.f(1) [[2, 2]] sage: mg.f(2) """
r""" Return `\varepsilon_i` of ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: mg = V.module_generators[0] sage: mg.epsilon(2) 0 sage: mg.f(1).epsilon(1) 1 """
r""" Return `\varphi_i` of ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: mg = V.module_generators[0] sage: mg.phi(1) 1 sage: mg.phi(2) 0 """
""" Return the weight of ``self``.
EXAMPLES::
sage: B = crystals.Tableaux(['B',3], shape=[1]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: psi = B.crystal_morphism(C.module_generators) sage: V = psi.image() sage: mg = V.module_generators[0] sage: mg.weight() (1, 0, 0) sage: mg.f(1).weight() (0, 1, 0) sage: all(V(psi(x)).weight() == x.weight() for x in B) True """ for i in self.index_set())
# TODO: implement a devirtualization map
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