Crystal of Rigged Configurations¶
AUTHORS:
- Travis Scrimshaw (2010-09-26): Initial version
We only consider the highest weight crystal structure, not the Kirillov-Reshetikhin structure, and we extend this to symmetrizable types.
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class
sage.combinat.rigged_configurations.rc_crystal.
CrystalOfNonSimplyLacedRC
(vct, wt, WLR)¶ Bases:
sage.combinat.rigged_configurations.rc_crystal.CrystalOfRiggedConfigurations
Highest weight crystal of rigged configurations in non-simply-laced type.
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Element
¶ alias of
sage.combinat.rigged_configurations.rigged_configuration_element.RCHWNonSimplyLacedElement
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from_virtual
(vrc)¶ Convert
vrc
in the virtual crystal into a rigged configuration of the original Cartan type.INPUT:
vrc
– a virtual rigged configuration
EXAMPLES:
sage: La = RootSystem(['C', 3]).weight_lattice().fundamental_weights() sage: vct = CartanType(['C', 3]).as_folding() sage: RC = crystals.RiggedConfigurations(vct, La[2]) sage: elt = RC(partition_list=[[0], [1], [1]]) sage: elt == RC.from_virtual(RC.to_virtual(elt)) True
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to_virtual
(rc)¶ Convert
rc
into a rigged configuration in the virtual crystal.INPUT:
rc
– a rigged configuration element
EXAMPLES:
sage: La = RootSystem(['C', 3]).weight_lattice().fundamental_weights() sage: vct = CartanType(['C', 3]).as_folding() sage: RC = crystals.RiggedConfigurations(vct, La[2]) sage: elt = RC(partition_list=[[], [1], [1]]); elt (/) 0[ ]0 -1[ ]-1 sage: RC.to_virtual(elt) (/) 0[ ]0 -2[ ][ ]-2 0[ ]0 (/)
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virtual
()¶ Return the corresponding virtual crystal.
EXAMPLES:
sage: La = RootSystem(['C', 2, 1]).weight_lattice().fundamental_weights() sage: vct = CartanType(['C', 2, 1]).as_folding() sage: RC = crystals.RiggedConfigurations(vct, La[0]) sage: RC Crystal of rigged configurations of type ['C', 2, 1] and weight Lambda[0] sage: RC.virtual Crystal of rigged configurations of type ['A', 3, 1] and weight 2*Lambda[0]
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class
sage.combinat.rigged_configurations.rc_crystal.
CrystalOfRiggedConfigurations
(wt, WLR)¶ Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
A highest weight crystal of rigged configurations.
The crystal structure for finite simply-laced types is given in [CrysStructSchilling06]. These were then shown to be the crystal operators in all finite types in [SS2015], all simply-laced and a large class of foldings of simply-laced types in [SS2015II], and all symmetrizable types (uniformly) in [SS2017].
INPUT:
cartan_type
– (optional) a Cartan type or a Cartan type given as a foldingwt
– the highest weight vector in the weight lattice
EXAMPLES:
For simplicity, we display the rigged configurations horizontally:
sage: RiggedConfigurations.options.display='horizontal'
We start with a simply-laced finite type:
sage: La = RootSystem(['A', 2]).weight_lattice().fundamental_weights() sage: RC = crystals.RiggedConfigurations(La[1] + La[2]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([1,2]) 0[ ]0 0[ ]-1 sage: mg.f_string([1,2,2]) 0[ ]0 -2[ ][ ]-2 sage: mg.f_string([1,2,2,2]) sage: mg.f_string([2,1,1,2]) -1[ ][ ]-1 -1[ ][ ]-1 sage: RC.cardinality() 8 sage: T = crystals.Tableaux(['A', 2], shape=[2,1]) sage: RC.digraph().is_isomorphic(T.digraph(), edge_labels=True) True
We construct a non-simply-laced affine type:
sage: La = RootSystem(['C', 3]).weight_lattice().fundamental_weights() sage: RC = crystals.RiggedConfigurations(La[2]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([2,3]) (/) 1[ ]1 -1[ ]-1 sage: T = crystals.Tableaux(['C', 3], shape=[1,1]) sage: RC.digraph().is_isomorphic(T.digraph(), edge_labels=True) True
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: RC = crystals.RiggedConfigurations(vct, La[2]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([2,3]) (/) 0[ ]0 -1[ ]-1 sage: T = crystals.Tableaux(['C', 3], shape=[1,1]) sage: RC.digraph().is_isomorphic(T.digraph(), edge_labels=True) True
We reset the global options:
sage: RiggedConfigurations.options._reset()
REFERENCES:
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Element
¶ alias of
sage.combinat.rigged_configurations.rigged_configuration_element.RCHighestWeightElement
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options
= Current options for RiggedConfigurations - convention: English - display: vertical - element_ascii_art: True - half_width_boxes_type_B: True¶
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weight_lattice_realization
()¶ Return the weight lattice realization used to express the weights of elements in
self
.EXAMPLES:
sage: La = RootSystem(['A', 2, 1]).weight_lattice(extended=True).fundamental_weights() sage: RC = crystals.RiggedConfigurations(La[0]) sage: RC.weight_lattice_realization() Extended weight lattice of the Root system of type ['A', 2, 1]