AUTHORS:
We only consider the highest weight crystal structure, not the Kirillov-Reshetikhin structure, and we extend this to symmetrizable types.
Bases: sage.combinat.rigged_configurations.rc_crystal.CrystalOfRiggedConfigurations
Highest weight crystal of rigged configurations in non-simply-laced type.
alias of RCHWNonSimplyLacedElement
Convert vrc in the virtual crystal into a rigged configution of the original Cartan type.
INPUT:
EXAMPLES:
sage: La = RootSystem(['C', 3]).weight_lattice().fundamental_weights()
sage: RC = crystals.RiggedConfigurations(La[2])
sage: elt = RC(partition_list=[[0], [1], [1]])
sage: elt == RC.from_virtual(RC.to_virtual(elt))
True
Convert rc into a rigged configuration in the virtual crystal.
INPUT:
EXAMPLES:
sage: La = RootSystem(['C', 3]).weight_lattice().fundamental_weights()
sage: RC = crystals.RiggedConfigurations(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
(/)
Return the corresponding virtual crystal.
EXAMPLES:
sage: La = RootSystem(['C', 2, 1]).weight_lattice().fundamental_weights()
sage: RC = crystals.RiggedConfigurations(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]
Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation
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 [SchScr] and all simply-laced and a large class of foldings of simply-laced types in [SalScr].
INPUT:
EXAMPLES:
For simplicity, we display the rigged configurations horizontally:
sage: RiggedConfigurations.global_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 reset the global options:
sage: RiggedConfigurations.global_options.reset()
REFERENCES:
[SchScr] | Anne Schilling and Travis Scrimshaw. Crystal structure on rigged configurations and the filling map. Arxiv 1409.2920. |
[SalScr] | Ben Salisbury and Travis Scrimshaw.
A rigged configuration model for ![]() |
alias of RCHighestWeightElement
Sets and displays the global options for rigged configurations. If no parameters are set, then the function returns a copy of the options dictionary.
The options to partitions can be accessed as the method RiggedConfigurations.global_options of RiggedConfigurations.
OPTIONS:
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.global_options(display="horizontal", convention="french")
sage: elt
-1[ ]-1 1[ ][ ][ ]1 -1[ ]-1
-3[ ][ ][ ]-3
Changing the convention for rigged configurations also changes the convention option for tableaux and vice versa:
sage: T = Tableau([[1,2,3],[4,5]])
sage: T.pp()
4 5
1 2 3
sage: Tableaux.global_options(convention="english")
sage: elt
-3[ ][ ][ ]-3 1[ ][ ][ ]1 -1[ ]-1
-1[ ]-1
sage: T.pp()
1 2 3
4 5
sage: RiggedConfigurations.global_options.reset()
See GlobalOptions for more features of these options.