Rigged Configurations of \(\mathcal{B}(\infty)\)

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 \(\mathcal{B}(\infty)\) in non-simply-laced types.

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

Bases: sage.combinat.rigged_configurations.rigged_configuration_element.RCNonSimplyLacedElement

A rigged configuration in \(\mathcal{B}(\infty)\) in non-simply-laced types.

weight()

Return the weight of self.

EXAMPLES:

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

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

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

to_virtual(rc)

Convert rc into a rigged configuration in the virtual crystal.

INPUT:

• rc – a rigged configuration element

EXAMPLES:

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

virtual()

Return the corresponding virtual crystal.

EXAMPLES:

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

Rigged configuration model for \(\mathcal{B}(\infty)\).

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 \(f_a\).

INPUT:

• cartan_type – a Cartan type

EXAMPLES:

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

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

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: VRC = crystals.infinity.RiggedConfigurations(vct)
sage: mg = VRC.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

sage: G = RC.subcrystal(max_depth=5).digraph()
sage: VG = VRC.subcrystal(max_depth=5).digraph()
sage: G.is_isomorphic(VG, edge_labels=True)
True

We can also construct \(B(\infty)\) 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   0[ ]1   0[ ]0    -4[ ][ ][ ]-2
             0[ ]0   0[ ]-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   0[ ]0    -4[ ][ ][ ]-2
0[ ]-1   0[ ]1   0[ ]-1

We reset the global options:

sage: RiggedConfigurations.options._reset()

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

Bases: sage.combinat.rigged_configurations.rigged_configuration_element.RiggedConfigurationElement

A rigged configuration in \(\mathcal{B}(\infty)\) in simply-laced types.

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()
-2*delta

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

-3[ ][ ][ ]-3
-1[ ]-1

1[ ][ ][ ]1

-1[ ]-1

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: RC = crystals.infinity.RiggedConfigurations(['A', 2, 1])
sage: RC.weight_lattice_realization()
Extended weight lattice of the Root system of type ['A', 2, 1]