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.

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.

Element

alias of RCHWNonSimplyLacedElement

from_virtual(vrc)

Convert vrc in the virtual crystal into a rigged configution of the original Cartan type.

INPUT:

  • vrc – a virtual rigged configuration

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
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: RC = crystals.RiggedConfigurations(La[2])
sage: elt = RC(partition_list=[[], [1], [1]]); elt
<BLANKLINE>
(/)
<BLANKLINE>
0[ ]0
<BLANKLINE>
-1[ ]-1
<BLANKLINE>
sage: RC.to_virtual(elt)
<BLANKLINE>
(/)
<BLANKLINE>
0[ ]0
<BLANKLINE>
-2[ ][ ]-2
<BLANKLINE>
0[ ]0
<BLANKLINE>
(/)
<BLANKLINE>
virtual()

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]
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 [SchScr] and all simply-laced and a large class of foldings of simply-laced types in [SalScr].

INPUT:

  • cartan_type – (optional) a Cartan type
  • wt – 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 reset the global options:

sage: RiggedConfigurations.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 \(B(\infty)\). Arxiv 1404.6539.
Element

alias of RCHighestWeightElement

global_options(*args, **kwds)

Deprecated: Use options() instead. See trac ticket #18555 for details.

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 options to partitions can be accessed as the method RiggedConfigurations.options of RiggedConfigurations.

OPTIONS:

  • convention – (default: English) Sets the convention used for displaying tableaux and partitions
    • English – use the English convention
    • French – use the French convention
  • display – (default: vertical) Specifies how rigged configurations should be printed
    • horizontal – displayed horizontally
    • vertical – displayed vertically
  • element_ascii_art – (default: True) display using the repr option element_ascii_art
  • half_width_boxes_type_B – (default: True) display the last rigged partition in affine type B as half width boxes
  • notation – alternative name for convention

EXAMPLES:

sage: RC = RiggedConfigurations(['A',3,1], [[2,2],[1,1],[1,1]])
sage: elt = RC(partition_list=[[3,1], [3], [1]])
sage: elt
<BLANKLINE>
-3[ ][ ][ ]-3
-1[ ]-1
<BLANKLINE>
1[ ][ ][ ]1
<BLANKLINE>
-1[ ]-1
<BLANKLINE>
sage: RiggedConfigurations.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.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 GlobalOptions for more features of these options.

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]