Rigged Configurations of

AUTHORS:

• Travis Scrimshaw (2013-04-16): Initial version

class sage.combinat.rigged_configurations.rc_infinity.InfinityCrystalOfNonSimplyLacedRC(vct)

Bases: sage.combinat.rigged_configurations.rc_infinity.InfinityCrystalOfRiggedConfigurations

Rigged configurations for in non-simply-laced types.

class Element(parent, rigged_partitions=[], **options)

Bases: sage.combinat.rigged_configurations.rigged_configuration_element.RCNonSimplyLacedElement

A rigged configuration in in non-simply-laced types.

TESTS:

sage: RC = crystals.infinity.RiggedConfigurations(['C', 3])
sage: elt = RC(partition_list=[[1],[1,1],[1]])
sage: TestSuite(elt).run()

weight()

Return the weight of self.

EXAMPLES:

sage: RC = crystals.infinity.RiggedConfigurations(['C', 3])
sage: elt = RC(partition_list=[[1],[1,1],[1]], rigging_list=[[0],[-1,-1],[0]])
sage: elt.weight()
(1, 1, 0)

sage: RC = crystals.infinity.RiggedConfigurations(['F', 4, 1])
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]
sage: al = RC.weight_lattice_realization().simple_roots()
sage: wt == al[0] + al[1] + 2*al[2] + al[3] + al[4]
True

InfinityCrystalOfNonSimplyLacedRC.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: RC = crystals.infinity.RiggedConfigurations(['C', 2])
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

InfinityCrystalOfNonSimplyLacedRC.to_virtual(rc)

Convert rc into a rigged configuration in the virtual crystal.

INPUT:

• rc – a rigged configuration element

EXAMPLES:

sage: RC = crystals.infinity.RiggedConfigurations(['C', 2])
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]

InfinityCrystalOfNonSimplyLacedRC.virtual()

Return the corresponding virtual crystal.

EXAMPLES:

sage: RC = crystals.infinity.RiggedConfigurations(['C', 3])
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.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

Rigged configuration model for .

The crystal is generated by the empty rigged configuration with the same crystal structure given by the highest weight model except we remove the condition that the resulting rigged configuration needs to be valid when applying .

INPUT:

• cartan_type – a Cartan type

EXAMPLES:

For simplicity, we display all of the rigged configurations horizontally:

sage: RiggedConfigurations.global_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.global_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   -2[ ]-1   0[ ]0
        -2[ ]-1
sage: mg.f_string([2,3,2,1,3,2])
-1[ ]-1  -2[ ][ ][ ]-1  -1[ ][ ]0

We can also construct 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  -1[ ]0  -1[ ]-1  -4[ ][ ][ ]-2
            -1[ ]0  -1[ ]-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  -1[ ]-1  -4[ ][ ][ ]-2
0[ ]-1  0[ ]1  -1[ ]-1

We reset the global options:

sage: RiggedConfigurations.global_options.reset()

class Element(parent, rigged_partitions=[], **options)

Bases: sage.combinat.rigged_configurations.rigged_configuration_element.RiggedConfigurationElement

A rigged configuration in in simply-laced types.

TESTS:

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: TestSuite(elt).run()

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()
0

InfinityCrystalOfRiggedConfigurations.global_options(*get_value, **set_value)

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:

• 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

-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.