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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sage.combinat.rigged_configurations.rc_crystal.
CrystalOfNonSimplyLacedRC
¶ Highest weight crystal of rigged configurations in non-simply-laced type.
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sage.combinat.rigged_configurations.rc_crystal.
CrystalOfRiggedConfigurations
¶ 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: