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r""" Rigged Configurations of `\mathcal{B}(\infty)`
AUTHORS:
- Travis Scrimshaw (2013-04-16): Initial version """
#***************************************************************************** # Copyright (C) 2013 Travis Scrimshaw <tscrim@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/ #*****************************************************************************
from sage.misc.lazy_attribute import lazy_attribute from sage.structure.unique_representation import UniqueRepresentation from sage.structure.parent import Parent from sage.categories.highest_weight_crystals import HighestWeightCrystals from sage.categories.homset import Hom from sage.combinat.root_system.cartan_type import CartanType from sage.combinat.rigged_configurations.rigged_configuration_element import ( RiggedConfigurationElement, RCNonSimplyLacedElement) from sage.combinat.rigged_configurations.rigged_configurations import RiggedConfigurations from sage.combinat.rigged_configurations.rigged_partition import RiggedPartition
# Note on implementation, this class is used for simply-laced types only class InfinityCrystalOfRiggedConfigurations(UniqueRepresentation, Parent): r""" Rigged configuration model for `\mathcal{B}(\infty)`.
The crystal is generated by the empty rigged configuration with the same crystal structure given by the :class:`highest weight model <sage.combinat.rigged_configurations.rc_crystal.CrystalOfRiggedConfigurations>` except we remove the condition that the resulting rigged configuration needs to be valid when applying `f_a`.
INPUT:
- ``cartan_type`` -- a Cartan type
EXAMPLES:
For simplicity, we display all of the rigged configurations horizontally::
sage: RiggedConfigurations.options(display='horizontal')
We begin with a simply-laced finite type::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3]); RC The infinity crystal of rigged configurations of type ['A', 3]
sage: RC.options(display='horizontal')
sage: mg = RC.highest_weight_vector(); mg (/) (/) (/) sage: elt = mg.f_string([2,1,3,2]); elt 0[ ]0 -2[ ]-1 0[ ]0 -2[ ]-1 sage: elt.e(1) sage: elt.e(3) sage: mg.f_string([2,1,3,2]).e(2) -1[ ]-1 0[ ]1 -1[ ]-1 sage: mg.f_string([2,3,2,1,3,2]) 0[ ]0 -3[ ][ ]-1 -1[ ][ ]-1 -2[ ]-1
Next we consider a non-simply-laced finite type::
sage: RC = crystals.infinity.RiggedConfigurations(['C', 3]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([2,1,3,2]) 0[ ]0 -1[ ]0 0[ ]0 -1[ ]-1 sage: mg.f_string([2,3,2,1,3,2]) 0[ ]-1 -1[ ][ ]-1 -1[ ][ ]0 -1[ ]0
We can construct rigged configurations using a diagram folding of a simply-laced type. This yields an equivalent but distinct crystal::
sage: vct = CartanType(['C', 3]).as_folding() sage: VRC = crystals.infinity.RiggedConfigurations(vct) sage: mg = VRC.highest_weight_vector() sage: mg.f_string([2,1,3,2]) 0[ ]0 -2[ ]-1 0[ ]0 -2[ ]-1 sage: mg.f_string([2,3,2,1,3,2]) -1[ ]-1 -2[ ][ ][ ]-1 -1[ ][ ]0
sage: G = RC.subcrystal(max_depth=5).digraph() sage: VG = VRC.subcrystal(max_depth=5).digraph() sage: G.is_isomorphic(VG, edge_labels=True) True
We can also construct `B(\infty)` using rigged configurations in affine types::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3, 1]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([0,1,2,3,0,1,3]) -1[ ]0 -1[ ]-1 1[ ]1 -1[ ][ ]-1 -1[ ]0 -1[ ]-1
sage: RC = crystals.infinity.RiggedConfigurations(['C', 3, 1]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([1,2,3,0,1,2,3,3,0]) -2[ ][ ]-1 0[ ]1 0[ ]0 -4[ ][ ][ ]-2 0[ ]0 0[ ]-1
sage: RC = crystals.infinity.RiggedConfigurations(['A', 6, 2]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([1,2,3,0,1,2,3,3,0]) 0[ ]-1 0[ ]1 0[ ]0 -4[ ][ ][ ]-2 0[ ]-1 0[ ]1 0[ ]-1
We reset the global options::
sage: RiggedConfigurations.options._reset() """ @staticmethod def __classcall_private__(cls, cartan_type): r""" Normalize the input arguments to ensure unique representation.
EXAMPLES::
sage: RC1 = crystals.infinity.RiggedConfigurations(CartanType(['A',3])) sage: RC2 = crystals.infinity.RiggedConfigurations(['A',3]) sage: RC2 is RC1 True """
def __init__(self, cartan_type): r""" Initialize ``self``.
EXAMPLES::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 2]) sage: TestSuite(RC).run() sage: RC = crystals.infinity.RiggedConfigurations(['A', 2, 1]) sage: TestSuite(RC).run() # long time sage: RC = crystals.infinity.RiggedConfigurations(['C', 2]) sage: TestSuite(RC).run() # long time sage: RC = crystals.infinity.RiggedConfigurations(['C', 2, 1]) sage: TestSuite(RC).run() # long time """ # We store the Cartan matrix for the vacancy number calculations for speed
options = RiggedConfigurations.options
def _repr_(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: crystals.infinity.RiggedConfigurations(['A', 3]) The infinity crystal of rigged configurations of type ['A', 3] """
def _element_constructor_(self, lst=None, **options): """ Construct an element of ``self`` from ``lst``.
EXAMPLES::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3, 1]) sage: ascii_art(RC(partition_list=[[1,1]]*4, rigging_list=[[1,1], [0,0], [0,0], [-1,-1]])) 0[ ]1 0[ ]0 0[ ]0 0[ ]-1 0[ ]1 0[ ]0 0[ ]0 0[ ]-1
sage: RC = crystals.infinity.RiggedConfigurations(['C', 3]) sage: ascii_art(RC(partition_list=[[1],[1,1],[1]], rigging_list=[[0],[0,-1],[0]])) 0[ ]0 -1[ ]0 0[ ]0 -1[ ]-1
TESTS:
Check that :trac:`17054` is fixed::
sage: RC = RiggedConfigurations(['A',2,1], [[1,1]]*4 + [[2,1]]*4) sage: B = crystals.infinity.RiggedConfigurations(['A',2]) sage: x = RC().f_string([2,2,1,1,2,1,2,1]) sage: ascii_art(x) 0[ ][ ][ ][ ]-4 0[ ][ ][ ][ ]0 sage: ascii_art(B(x)) -4[ ][ ][ ][ ]-4 -4[ ][ ][ ][ ]0 sage: x == RC().f_string([2,2,1,1,2,1,2,1]) True """ lst = [p._clone() for p in lst] # Make a deep copy
def _coerce_map_from_(self, P): """ Return ``True`` or the coerce map from ``P`` if a map exists.
EXAMPLES::
sage: T = crystals.infinity.Tableaux(['A',3]) sage: RC = crystals.infinity.RiggedConfigurations(['A',3]) sage: RC._coerce_map_from_(T) Crystal Isomorphism morphism: From: The infinity crystal of tableaux of type ['A', 3] To: The infinity crystal of rigged configurations of type ['A', 3] """ and self.cartan_type().is_simply_laced()):
def _calc_vacancy_number(self, partitions, a, i, **options): r""" Calculate the vacancy number `p_i^{(a)}(\nu)` in ``self``.
This assumes that `\gamma_a = 1` for all `a` and `(\alpha_a \mid \alpha_b ) = A_{ab}`.
INPUT:
- ``partitions`` -- the list of rigged partitions we are using
- ``a`` -- the rigged partition index
- ``i`` -- the row length
TESTS::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 4]) sage: elt = RC(partition_list=[[1], [1], [], []]) sage: RC._calc_vacancy_number(elt.nu(), 0, 1) -1 """ for b,nu in enumerate(partitions))
for b,nu in enumerate(partitions))
# FIXME: Remove this method!!! def weight_lattice_realization(self): """ Return the weight lattice realization used to express the weights of elements in ``self``.
EXAMPLES::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 2, 1]) sage: RC.weight_lattice_realization() Extended weight lattice of the Root system of type ['A', 2, 1] """ return R.weight_lattice()
class Element(RiggedConfigurationElement): """ A rigged configuration in `\mathcal{B}(\infty)` in simply-laced types.
TESTS::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3, 1]) sage: elt = RC(partition_list=[[1,1]]*4, rigging_list=[[1,1], [0,0], [0,0], [-1,-1]]) sage: TestSuite(elt).run() """ def weight(self): """ Return the weight of ``self``.
EXAMPLES::
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3, 1]) sage: elt = RC(partition_list=[[1,1]]*4, rigging_list=[[1,1], [0,0], [0,0], [-1,-1]]) sage: elt.weight() -2*delta """
class InfinityCrystalOfNonSimplyLacedRC(InfinityCrystalOfRiggedConfigurations): r""" Rigged configurations for `\mathcal{B}(\infty)` in non-simply-laced types. """ def __init__(self, vct): """ Initialize ``self``.
EXAMPLES::
sage: vct = CartanType(['C', 2]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: TestSuite(RC).run() # long time sage: vct = CartanType(['C', 2, 1]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: TestSuite(RC).run() # long time """
def _coerce_map_from_(self, P): """ Return ``True`` or the coerce map from ``P`` if a map exists.
EXAMPLES::
sage: T = crystals.infinity.Tableaux(['C',3]) sage: vct = CartanType(['C',3]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: RC._coerce_map_from_(T) Crystal Isomorphism morphism: From: The infinity crystal of tableaux of type ['C', 3] To: The infinity crystal of rigged configurations of type ['C', 3] """
def _calc_vacancy_number(self, partitions, a, i): r""" Calculate the vacancy number `p_i^{(a)}(\nu)` in ``self``.
INPUT:
- ``partitions`` -- the list of rigged partitions we are using
- ``a`` -- the rigged partition index
- ``i`` -- the row length
TESTS::
sage: La = RootSystem(['C', 2]).weight_lattice().fundamental_weights() sage: vct = CartanType(['C', 2]).as_folding() sage: RC = crystals.RiggedConfigurations(vct, La[1]) sage: elt = RC(partition_list=[[1], [1]]) sage: RC._calc_vacancy_number(elt.nu(), 0, 1) 0 sage: RC._calc_vacancy_number(elt.nu(), 1, 1) -1 """
for b,nu in enumerate(partitions))
@lazy_attribute def virtual(self): """ Return the corresponding virtual crystal.
EXAMPLES::
sage: vct = CartanType(['C', 3]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: RC The infinity crystal of rigged configurations of type ['C', 3] sage: RC.virtual The infinity crystal of rigged configurations of type ['A', 5] """
def to_virtual(self, rc): """ Convert ``rc`` into a rigged configuration in the virtual crystal.
INPUT:
- ``rc`` -- a rigged configuration element
EXAMPLES::
sage: vct = CartanType(['C', 2]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: mg = RC.highest_weight_vector() sage: elt = mg.f_string([1,2,2,1,1]); elt <BLANKLINE> -3[ ][ ][ ]-2 <BLANKLINE> -1[ ][ ]0 <BLANKLINE> sage: velt = RC.to_virtual(elt); velt <BLANKLINE> -3[ ][ ][ ]-2 <BLANKLINE> -2[ ][ ][ ][ ]0 <BLANKLINE> -3[ ][ ][ ]-2 <BLANKLINE> sage: velt.parent() The infinity crystal of rigged configurations of type ['A', 3] """ rigging_list=riggings)
def from_virtual(self, vrc): """ Convert ``vrc`` in the virtual crystal into a rigged configuration of the original Cartan type.
INPUT:
- ``vrc`` -- a virtual rigged configuration
EXAMPLES::
sage: vct = CartanType(['C', 2]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: elt = RC(partition_list=[[3],[2]], rigging_list=[[-2],[0]]) sage: vrc_elt = RC.to_virtual(elt) sage: ret = RC.from_virtual(vrc_elt); ret <BLANKLINE> -3[ ][ ][ ]-2 <BLANKLINE> -1[ ][ ]0 <BLANKLINE> sage: ret == elt True """ # TODO: Handle special cases for A^{(2)} even and its dual? rigging_list=riggings)
class Element(RCNonSimplyLacedElement): """ A rigged configuration in `\mathcal{B}(\infty)` in non-simply-laced types.
TESTS::
sage: vct = CartanType(['C', 3]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: elt = RC(partition_list=[[1],[1,1],[1]]) sage: TestSuite(elt).run() """ def weight(self): """ Return the weight of ``self``.
EXAMPLES::
sage: vct = CartanType(['C', 3]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: elt = RC(partition_list=[[1],[1,1],[1]], rigging_list=[[0],[-1,-1],[0]]) sage: elt.weight() (-1, -1, 0)
sage: vct = CartanType(['F', 4, 1]).as_folding() sage: RC = crystals.infinity.RiggedConfigurations(vct) sage: mg = RC.highest_weight_vector() sage: elt = mg.f_string([1,0,3,4,2,2]); ascii_art(elt) -1[ ]-1 0[ ]1 -2[ ][ ]-2 0[ ]1 -1[ ]-1 sage: wt = elt.weight(); wt -Lambda[0] + Lambda[1] - 2*Lambda[2] + 3*Lambda[3] - Lambda[4] - delta sage: al = RC.weight_lattice_realization().simple_roots() sage: wt == -(al[0] + al[1] + 2*al[2] + al[3] + al[4]) True """
# deprecations from trac:18555 from sage.misc.superseded import deprecated_function_alias InfinityCrystalOfRiggedConfigurations.global_options = deprecated_function_alias(18555, InfinityCrystalOfRiggedConfigurations.options)
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