Bijection between rigged configurations for \(B(\infty)\) and marginally large tableaux - Combinatorics

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# Bijection between rigged configurations for \(B(\infty)\) and marginally large tableaux

AUTHORS:

- Travis Scrimshaw (2015-07-01): Initial version

REFERENCES:

[RC-MLT]

Ben Salisbury and Travis Scrimshaw. Connecting marginally large tableaux and rigged configurations via crystals. Preprint. arXiv 1505.07040.

class sage.combinat.rigged_configurations.bij_infinity.FromRCIsomorphism

  Bases: `Morphism`

  Crystal isomorphism of \(B(\infty)\) in the rigged configuration model to the tableau model.

class sage.combinat.rigged_configurations.bij_infinity.FromTableauIsomorphism

  Bases: `Morphism`

  Crystal isomorphism of \(B(\infty)\) in the tableau model to the rigged configuration model.

class sage.combinat.rigged_configurations.bij_infinity.MLTToRCBijectionTypeB(tp_krt)

  Bases: `KRTToRCBijectionTypeB`

  run()

    Run the bijection from a marginally large tableaux to a rigged configuration.

    EXAMPLES:

    Sage

        sage: vct = CartanType(['B',4]).as_folding()
        sage: RC = crystals.infinity.RiggedConfigurations(vct)
        sage: T = crystals.infinity.Tableaux(['B',4])
        sage: Psi = T.crystal_morphism({T.module_generators[0]: RC.module_generators[0]})
        sage: TS = [x.value for x in T.subcrystal(max_depth=4)]
        sage: all(Psi(b) == RC(b) for b in TS) # long time # indirect doctest
        True

    Python

        >>> from sage.all import *
        >>> vct = CartanType(['B',Integer(4)]).as_folding()
        >>> RC = crystals.infinity.RiggedConfigurations(vct)
        >>> T = crystals.infinity.Tableaux(['B',Integer(4)])
        >>> Psi = T.crystal_morphism({T.module_generators[Integer(0)]: RC.module_generators[Integer(0)]})
        >>> TS = [x.value for x in T.subcrystal(max_depth=Integer(4))]
        >>> all(Psi(b) == RC(b) for b in TS) # long time # indirect doctest
        True

class sage.combinat.rigged_configurations.bij_infinity.MLTToRCBijectionTypeD(tp_krt)

  Bases: `KRTToRCBijectionTypeD`

  run()

    Run the bijection from a marginally large tableaux to a rigged configuration.

    EXAMPLES:

    Sage

        sage: RC = crystals.infinity.RiggedConfigurations(['D',4])
        sage: T = crystals.infinity.Tableaux(['D',4])
        sage: Psi = T.crystal_morphism({T.module_generators[0]: RC.module_generators[0]})
        sage: TS = [x.value for x in T.subcrystal(max_depth=4)]
        sage: all(Psi(b) == RC(b) for b in TS) # long time # indirect doctest
        True

    Python

        >>> from sage.all import *
        >>> RC = crystals.infinity.RiggedConfigurations(['D',Integer(4)])
        >>> T = crystals.infinity.Tableaux(['D',Integer(4)])
        >>> Psi = T.crystal_morphism({T.module_generators[Integer(0)]: RC.module_generators[Integer(0)]})
        >>> TS = [x.value for x in T.subcrystal(max_depth=Integer(4))]
        >>> all(Psi(b) == RC(b) for b in TS) # long time # indirect doctest
        True

class sage.combinat.rigged_configurations.bij_infinity.RCToMLTBijectionTypeB(RC_element)

  Bases: `RCToKRTBijectionTypeB`

  run()

    Run the bijection from rigged configurations to a marginally large tableau.

    EXAMPLES:

    Sage

        sage: vct = CartanType(['B',4]).as_folding()
        sage: RC = crystals.infinity.RiggedConfigurations(vct)
        sage: T = crystals.infinity.Tableaux(['B',4])
        sage: Psi = RC.crystal_morphism({RC.module_generators[0]: T.module_generators[0]})
        sage: RCS = [x.value for x in RC.subcrystal(max_depth=4)]
        sage: all(Psi(nu) == T(nu) for nu in RCS) # long time # indirect doctest
        True

    Python

        >>> from sage.all import *
        >>> vct = CartanType(['B',Integer(4)]).as_folding()
        >>> RC = crystals.infinity.RiggedConfigurations(vct)
        >>> T = crystals.infinity.Tableaux(['B',Integer(4)])
        >>> Psi = RC.crystal_morphism({RC.module_generators[Integer(0)]: T.module_generators[Integer(0)]})
        >>> RCS = [x.value for x in RC.subcrystal(max_depth=Integer(4))]
        >>> all(Psi(nu) == T(nu) for nu in RCS) # long time # indirect doctest
        True

class sage.combinat.rigged_configurations.bij_infinity.RCToMLTBijectionTypeD(RC_element)

  Bases: `RCToKRTBijectionTypeD`

  run()

    Run the bijection from rigged configurations to a marginally large tableau.

    EXAMPLES:

    Sage

        sage: RC = crystals.infinity.RiggedConfigurations(['D',4])
        sage: T = crystals.infinity.Tableaux(['D',4])
        sage: Psi = RC.crystal_morphism({RC.module_generators[0]: T.module_generators[0]})
        sage: RCS = [x.value for x in RC.subcrystal(max_depth=4)]
        sage: all(Psi(nu) == T(nu) for nu in RCS) # long time # indirect doctest
        True

    Python

        >>> from sage.all import *
        >>> RC = crystals.infinity.RiggedConfigurations(['D',Integer(4)])
        >>> T = crystals.infinity.Tableaux(['D',Integer(4)])
        >>> Psi = RC.crystal_morphism({RC.module_generators[Integer(0)]: T.module_generators[Integer(0)]})
        >>> RCS = [x.value for x in RC.subcrystal(max_depth=Integer(4))]
        >>> all(Psi(nu) == T(nu) for nu in RCS) # long time # indirect doctest
        True