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# 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)

  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)

  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:

  - `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.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 \(B(\infty)\). Arxiv 1404.6539.

  `Element`

    alias of `RCHighestWeightElement`

  `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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