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# 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.

    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
        <BLANKLINE>
        -3[ ][ ][ ]-2
        <BLANKLINE>
        -1[ ][ ]0
        <BLANKLINE>
        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
        <BLANKLINE>
        -3[ ][ ][ ]-2
        <BLANKLINE>
        -1[ ][ ]0
        <BLANKLINE>
        sage: velt = RC.to_virtual(elt); velt
        <BLANKLINE>
        -3[ ][ ][ ]-2
        <BLANKLINE>
        -2[ ][ ][ ][ ]0
        <BLANKLINE>
        -3[ ][ ][ ]-2
        <BLANKLINE>
        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 \(\mathcal{B}(\infty)\).

  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.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 \(B(\infty)\) 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  -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 \(\mathcal{B}(\infty)\) 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
        <BLANKLINE>
        -3[ ][ ][ ]-3
        -1[ ]-1
        <BLANKLINE>
        1[ ][ ][ ]1
        <BLANKLINE>
        -1[ ]-1
        <BLANKLINE>
        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.

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