Rigged Configurations of \(\mathcal{B}(\infty)\)¶
AUTHORS:
Travis Scrimshaw (2013-04-16): Initial version
- class sage.combinat.rigged_configurations.rc_infinity.InfinityCrystalOfNonSimplyLacedRC(vct)¶
Bases:
sage.combinat.rigged_configurations.rc_infinity.InfinityCrystalOfRiggedConfigurationsRigged configurations for \(\mathcal{B}(\infty)\) in non-simply-laced types.
- class Element(parent, rigged_partitions=[], **options)¶
Bases:
sage.combinat.rigged_configurations.rigged_configuration_element.RCNonSimplyLacedElementA rigged configuration in \(\mathcal{B}(\infty)\) in non-simply-laced types.
- weight()¶
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
- from_virtual(vrc)¶
Convert
vrcin 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 -3[ ][ ][ ]-2 -1[ ][ ]0 sage: ret == elt True
- to_virtual(rc)¶
Convert
rcinto 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 -3[ ][ ][ ]-2 -1[ ][ ]0 sage: velt = RC.to_virtual(elt); velt -3[ ][ ][ ]-2 -2[ ][ ][ ][ ]0 -3[ ][ ][ ]-2 sage: velt.parent() The infinity crystal of rigged configurations of type ['A', 3]
- virtual()¶
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]
- class sage.combinat.rigged_configurations.rc_infinity.InfinityCrystalOfRiggedConfigurations(cartan_type)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation,sage.structure.parent.ParentRigged configuration model for \(\mathcal{B}(\infty)\).
The crystal is generated by the empty rigged configuration with the same crystal structure given by the
highest weight modelexcept 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()
- class Element(parent, rigged_partitions=[], **options)¶
Bases:
sage.combinat.rigged_configurations.rigged_configuration_element.RiggedConfigurationElementA rigged configuration in \(\mathcal{B}(\infty)\) in simply-laced types.
- weight()¶
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
- options(*get_value, **set_value)¶
Sets and displays the options for rigged configurations. If no parameters are set, then the function returns a copy of the options dictionary.
The
optionsto partitions can be accessed as the methodRiggedConfigurations.optionsofRiggedConfigurations.OPTIONS:
convention– (default:English) Sets the convention used for displaying tableaux and partitionsEnglish– use the English conventionFrench– use the French convention
display– (default:vertical) Specifies how rigged configurations should be printedhorizontal– displayed horizontallyvertical– displayed vertically
element_ascii_art– (default:True) display using the repr optionelement_ascii_arthalf_width_boxes_type_B– (default:True) display the last rigged partition in affine type B as half width boxesnotation– alternative name forconvention
EXAMPLES:
sage: RC = RiggedConfigurations(['A',3,1], [[2,2],[1,1],[1,1]]) sage: elt = RC(partition_list=[[3,1], [3], [1]]) sage: elt -3[ ][ ][ ]-3 -1[ ]-1 1[ ][ ][ ]1 -1[ ]-1 sage: RiggedConfigurations.options(display="horizontal", convention="french") sage: elt -1[ ]-1 1[ ][ ][ ]1 -1[ ]-1 -3[ ][ ][ ]-3
Changing the
conventionfor rigged configurations also changes theconventionoption for tableaux and vice versa:sage: T = Tableau([[1,2,3],[4,5]]) sage: T.pp() 4 5 1 2 3 sage: Tableaux.options.convention="english" sage: elt -3[ ][ ][ ]-3 1[ ][ ][ ]1 -1[ ]-1 -1[ ]-1 sage: T.pp() 1 2 3 4 5 sage: RiggedConfigurations.options._reset()
See
GlobalOptionsfor more features of these options.
- weight_lattice_realization()¶
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]