Crystals of generalized Young walls - Combinatorics

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# Crystals of generalized Young walls

Generalized Young walls are certain generalizations of Young tableaux introduced in [KS2010] and designed to be a realization of the crystals \(\mathcal{B}(\infty)\) and \(\mathcal{B}(\lambda)\) in type \(A_n^{(1)}\).

REFERENCES:

- [KLRS2016]

- [KS2010]

AUTHORS:

- Lucas David-Roesler: initial version

- Ben Salisbury: initial version

- Travis Scrimshaw: initial version

class sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWalls(n, La)

  Bases: `InfinityCrystalOfGeneralizedYoungWalls`

  The crystal \(\mathcal{Y}(\lambda)\) of generalized Young walls of the given type with highest weight \(\lambda\).

  These were characterized in Theorem 4.1 of [KS2010]. See `GeneralizedYoungWall.in_highest_weight_crystal()`.

  INPUT:

  - `n` – type \(A_n^{(1)}\)

  - `weight` – dominant integral weight

  EXAMPLES:

  Sage

      sage: La = RootSystem(['A',3,1]).weight_lattice(extended=True).fundamental_weights()[1]
      sage: YLa = crystals.GeneralizedYoungWalls(3,La)
      sage: y = YLa([[0],[1,0,3,2,1],[2,1,0],[3]])
      sage: y.pp()
              3|
          0|1|2|
      1|2|3|0|1|
              0|
      sage: y.weight()
      -Lambda[0] + Lambda[2] + Lambda[3] - 3*delta
      sage: y.in_highest_weight_crystal(La)
      True
      sage: y.f(1)
      [[0], [1, 0, 3, 2, 1], [2, 1, 0], [3], [], [1]]
      sage: y.f(1).f(1)
      sage: yy = crystals.infinity.GeneralizedYoungWalls(3)([[0], [1, 0, 3, 2, 1], [2, 1, 0], [3], [], [1]])
      sage: yy.f(1)
      [[0], [1, 0, 3, 2, 1], [2, 1, 0], [3], [], [1], [], [], [], [1]]
      sage: yyy = yy.f(1)
      sage: yyy.in_highest_weight_crystal(La)
      False

      sage: LS = crystals.LSPaths(['A',3,1],[1,0,0,0])
      sage: C = LS.subcrystal(max_depth=4)
      sage: G = LS.digraph(subset=C)
      sage: P = RootSystem(['A',3,1]).weight_lattice(extended=True)
      sage: La = P.fundamental_weights()
      sage: YW = crystals.GeneralizedYoungWalls(3,La[0])
      sage: CW = YW.subcrystal(max_depth=4)
      sage: GW = YW.digraph(subset=CW)
      sage: GW.is_isomorphic(G,edge_labels=True)
      True

  Python

      >>> from sage.all import *
      >>> La = RootSystem(['A',Integer(3),Integer(1)]).weight_lattice(extended=True).fundamental_weights()[Integer(1)]
      >>> YLa = crystals.GeneralizedYoungWalls(Integer(3),La)
      >>> y = YLa([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2),Integer(1)],[Integer(2),Integer(1),Integer(0)],[Integer(3)]])
      >>> y.pp()
              3|
          0|1|2|
      1|2|3|0|1|
              0|
      >>> y.weight()
      -Lambda[0] + Lambda[2] + Lambda[3] - 3*delta
      >>> y.in_highest_weight_crystal(La)
      True
      >>> y.f(Integer(1))
      [[0], [1, 0, 3, 2, 1], [2, 1, 0], [3], [], [1]]
      >>> y.f(Integer(1)).f(Integer(1))
      >>> yy = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[Integer(0)], [Integer(1), Integer(0), Integer(3), Integer(2), Integer(1)], [Integer(2), Integer(1), Integer(0)], [Integer(3)], [], [Integer(1)]])
      >>> yy.f(Integer(1))
      [[0], [1, 0, 3, 2, 1], [2, 1, 0], [3], [], [1], [], [], [], [1]]
      >>> yyy = yy.f(Integer(1))
      >>> yyy.in_highest_weight_crystal(La)
      False

      >>> LS = crystals.LSPaths(['A',Integer(3),Integer(1)],[Integer(1),Integer(0),Integer(0),Integer(0)])
      >>> C = LS.subcrystal(max_depth=Integer(4))
      >>> G = LS.digraph(subset=C)
      >>> P = RootSystem(['A',Integer(3),Integer(1)]).weight_lattice(extended=True)
      >>> La = P.fundamental_weights()
      >>> YW = crystals.GeneralizedYoungWalls(Integer(3),La[Integer(0)])
      >>> CW = YW.subcrystal(max_depth=Integer(4))
      >>> GW = YW.digraph(subset=CW)
      >>> GW.is_isomorphic(G,edge_labels=True)
      True

  To display the crystal down to a specified depth:

  Sage

      sage: S = YLa.subcrystal(max_depth=4)
      sage: G = YLa.digraph(subset=S)
      sage: view(G) # not tested

  [Python]

      >>> from sage.all import *
      >>> S = YLa.subcrystal(max_depth=Integer(4))
      >>> G = YLa.digraph(subset=S)
      >>> view(G) # not tested

  Element

    alias of `CrystalOfGeneralizedYoungWallsElement`

class sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWallsElement(parent, data)

  Bases: `GeneralizedYoungWall`

  Element of the highest weight crystal of generalized Young walls.

  e(i)

    Compute the action of \(e_i\) restricted to the highest weight crystal.

    EXAMPLES:

    Sage

        sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()[1]
        sage: hwy = crystals.GeneralizedYoungWalls(2,La)([[],[1,0],[2,1]])
        sage: hwy.e(1)
        [[], [1, 0], [2]]
        sage: hwy.e(2)
        sage: hwy.e(3)

    Python

        >>> from sage.all import *
        >>> La = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=True).fundamental_weights()[Integer(1)]
        >>> hwy = crystals.GeneralizedYoungWalls(Integer(2),La)([[],[Integer(1),Integer(0)],[Integer(2),Integer(1)]])
        >>> hwy.e(Integer(1))
        [[], [1, 0], [2]]
        >>> hwy.e(Integer(2))
        >>> hwy.e(Integer(3))

  f(i)

    Compute the action of \(f_i\) restricted to the highest weight crystal.

    EXAMPLES:

    Sage

        sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()[1]
        sage: GYW = crystals.infinity.GeneralizedYoungWalls(2)
        sage: y = GYW([[],[1,0],[2,1]])
        sage: y.f(1)
        [[], [1, 0], [2, 1], [], [1]]
        sage: hwy = crystals.GeneralizedYoungWalls(2,La)([[],[1,0],[2,1]])
        sage: hwy.f(1)

    Python

        >>> from sage.all import *
        >>> La = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=True).fundamental_weights()[Integer(1)]
        >>> GYW = crystals.infinity.GeneralizedYoungWalls(Integer(2))
        >>> y = GYW([[],[Integer(1),Integer(0)],[Integer(2),Integer(1)]])
        >>> y.f(Integer(1))
        [[], [1, 0], [2, 1], [], [1]]
        >>> hwy = crystals.GeneralizedYoungWalls(Integer(2),La)([[],[Integer(1),Integer(0)],[Integer(2),Integer(1)]])
        >>> hwy.f(Integer(1))

  phi(i)

    Return the value \(\varepsilon_i(Y) + \langle h_i, \mathrm{wt}(Y)\rangle\), where \(h_i\) is the \(i\)-th simple coroot and \(Y\) is `self`.

    EXAMPLES:

    Sage

        sage: La = RootSystem(['A',3,1]).weight_lattice(extended=True).fundamental_weights()
        sage: y = crystals.GeneralizedYoungWalls(3,La[0])([])
        sage: y.phi(1)
        0
        sage: y.phi(2)
        0

    Python

        >>> from sage.all import *
        >>> La = RootSystem(['A',Integer(3),Integer(1)]).weight_lattice(extended=True).fundamental_weights()
        >>> y = crystals.GeneralizedYoungWalls(Integer(3),La[Integer(0)])([])
        >>> y.phi(Integer(1))
        0
        >>> y.phi(Integer(2))
        0

  weight()

    Return the weight of `self` in the highest weight crystal as an element of the weight lattice \(\bigoplus_{i=0}^n \ZZ \Lambda_i\).

    EXAMPLES:

    Sage

        sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()[1]
        sage: hwy = crystals.GeneralizedYoungWalls(2,La)([[],[1,0],[2,1]])
        sage: hwy.weight()
        Lambda[0] - Lambda[1] + Lambda[2] - delta

    Python

        >>> from sage.all import *
        >>> La = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=True).fundamental_weights()[Integer(1)]
        >>> hwy = crystals.GeneralizedYoungWalls(Integer(2),La)([[],[Integer(1),Integer(0)],[Integer(2),Integer(1)]])
        >>> hwy.weight()
        Lambda[0] - Lambda[1] + Lambda[2] - delta

class sage.combinat.crystals.generalized_young_walls.GeneralizedYoungWall(parent, data)

  Bases: `CombinatorialElement`

  A generalized Young wall.

  For more information, see `InfinityCrystalOfGeneralizedYoungWalls`.

  EXAMPLES:

  Sage

      sage: Y = crystals.infinity.GeneralizedYoungWalls(4)
      sage: mg = Y.module_generators[0]; mg.pp()
      0
      sage: mg.f_string([1,2,0,1]).pp()
      1|2|
      0|1|
         |

  Python

      >>> from sage.all import *
      >>> Y = crystals.infinity.GeneralizedYoungWalls(Integer(4))
      >>> mg = Y.module_generators[Integer(0)]; mg.pp()
      0
      >>> mg.f_string([Integer(1),Integer(2),Integer(0),Integer(1)]).pp()
      1|2|
      0|1|
         |

  Epsilon()

    Return \(\sum_{i=0}^n \varepsilon_i(Y) \Lambda_i\) where \(Y\) is `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(3)([[0],[1,0,3,2],[2,1],[3,2,1,0,3,2],[0],[],[2]])
        sage: y.Epsilon()
        Lambda[0] + 3*Lambda[2]

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(0)],[],[Integer(2)]])
        >>> y.Epsilon()
        Lambda[0] + 3*Lambda[2]

  Phi()

    Return \(\sum_{i=0}^n \varphi_i(Y) \Lambda_i\) where \(Y\) is `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(3)([[0],[1,0,3,2],[2,1],[3,2,1,0,3,2],[0],[],[2]])
        sage: y.Phi()
        -Lambda[0] + 3*Lambda[1] - Lambda[2] + 3*Lambda[3]

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.Phi()
        2*Lambda[0] + Lambda[1] - Lambda[2] + Lambda[3]

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(0)],[],[Integer(2)]])
        >>> y.Phi()
        -Lambda[0] + 3*Lambda[1] - Lambda[2] + 3*Lambda[3]

        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.Phi()
        2*Lambda[0] + Lambda[1] - Lambda[2] + Lambda[3]

  a(i, k)

    Return the number \(a_i(k)\) of \(i\)-colored boxes in the `k`-th column of `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(3)([[0],[1,0,3,2],[2,1],[3,2,1,0,3,2],[0],[],[2]])
        sage: y.a(1,2)
        1
        sage: y.a(0,2)
        1
        sage: y.a(3,2)
        0

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(0)],[],[Integer(2)]])
        >>> y.a(Integer(1),Integer(2))
        1
        >>> y.a(Integer(0),Integer(2))
        1
        >>> y.a(Integer(3),Integer(2))
        0

  column(k)

    Return the list of boxes from the `k`-th column of `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(3)([[0],[1,0,3,2],[2,1],[3,2,1,0,3,2],[0],[],[2]])
        sage: y.column(2)
        [None, 0, 1, 2, None, None, None]

        sage: hw = crystals.infinity.GeneralizedYoungWalls(5)([])
        sage: hw.column(1)
        []

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(0)],[],[Integer(2)]])
        >>> y.column(Integer(2))
        [None, 0, 1, 2, None, None, None]

        >>> hw = crystals.infinity.GeneralizedYoungWalls(Integer(5))([])
        >>> hw.column(Integer(1))
        []

  content()

    Return total number of blocks in `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(2)([[0],[1,0],[2,1,0,2],[],[1]])
        sage: y.content()
        8

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.content()
        13

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(2))([[Integer(0)],[Integer(1),Integer(0)],[Integer(2),Integer(1),Integer(0),Integer(2)],[],[Integer(1)]])
        >>> y.content()
        8

        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.content()
        13

  e(i)

    Return the application of the Kashiwara raising operator \(e_i\) on `self`.

    This will remove the \(i\)-colored box corresponding to the rightmost \(+\) in `self.signature(i)`.

    EXAMPLES:

    Sage

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.e(2)
        [[], [1, 0, 3, 2], [2, 1], [3, 2, 1, 0, 3, 2]]
        sage: _.e(2)
        [[], [1, 0, 3], [2, 1], [3, 2, 1, 0, 3, 2]]
        sage: _.e(2)
        [[], [1, 0, 3], [2, 1], [3, 2, 1, 0, 3]]
        sage: _.e(2)

    Python

        >>> from sage.all import *
        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.e(Integer(2))
        [[], [1, 0, 3, 2], [2, 1], [3, 2, 1, 0, 3, 2]]
        >>> _.e(Integer(2))
        [[], [1, 0, 3], [2, 1], [3, 2, 1, 0, 3, 2]]
        >>> _.e(Integer(2))
        [[], [1, 0, 3], [2, 1], [3, 2, 1, 0, 3]]
        >>> _.e(Integer(2))

  epsilon(i)

    Return the number of \(i\)-colored arrows in the \(i\)-string above `self` in the crystal graph.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: y.epsilon(1)
        0
        sage: y.epsilon(2)
        3
        sage: y.epsilon(0)
        0

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> y.epsilon(Integer(1))
        0
        >>> y.epsilon(Integer(2))
        3
        >>> y.epsilon(Integer(0))
        0

  f(i)

    Return the application of the Kashiwara lowering operator \(f_i\) on `self`.

    This will add an \(i\)-colored colored box to the site corresponding to the leftmost plus in `self.signature(i)`.

    EXAMPLES:

    Sage

        sage: hw = crystals.infinity.GeneralizedYoungWalls(2)([])
        sage: hw.f(1)
        [[], [1]]
        sage: _.f(2)
        [[], [1], [2]]
        sage: _.f(0)
        [[], [1, 0], [2]]
        sage: _.f(0)
        [[0], [1, 0], [2]]

    Python

        >>> from sage.all import *
        >>> hw = crystals.infinity.GeneralizedYoungWalls(Integer(2))([])
        >>> hw.f(Integer(1))
        [[], [1]]
        >>> _.f(Integer(2))
        [[], [1], [2]]
        >>> _.f(Integer(0))
        [[], [1, 0], [2]]
        >>> _.f(Integer(0))
        [[0], [1, 0], [2]]

  generate_signature(i)

    The \(i\)-signature of `self` (with whitespace where cancellation occurs) together with the unreduced sequence from \(\{+,-\}\). The result also records to the row and column position of the sign.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(2)([[0],[1,0],[2,1,0,2],[],[1]])
        sage: y.generate_signature(1)
        ([['+', 2, 5], ['-', 4, 1]], '  ')

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(2))([[Integer(0)],[Integer(1),Integer(0)],[Integer(2),Integer(1),Integer(0),Integer(2)],[],[Integer(1)]])
        >>> y.generate_signature(Integer(1))
        ([['+', 2, 5], ['-', 4, 1]], '  ')

  in_highest_weight_crystal(La)

    Return a boolean indicating if the generalized Young wall element is in the highest weight crystal cut out by the given highest weight `La`.

    By Theorem 4.1 of [KS2010], a generalized Young wall \(Y\) represents a vertex in the highest weight crystal \(Y(\lambda)\), with \(\lambda = \Lambda_{i_1} + \Lambda_{i_2} + \cdots + \Lambda_{i_\ell}\) a dominant integral weight of level \(\ell > 0\), if it satisfies the following condition. For each positive integer \(k\), if there exists \(j \in I\) such that \(a_j(k) - a_{j-1}(k) > 0\), then for some \(p = 1, \ldots, \ell\),

    \[j + k \equiv i_p + 1 \bmod n+1 \text{ and } a_j(k) - a_{j-1}(k) \le \lambda(h_{i_p}),\]

    where \(\{h_0, h_1, \ldots, h_n\}\) is the set of simple coroots attached to \(A_n^{(1)}\).

    EXAMPLES:

    Sage

        sage: La = RootSystem(['A',2,1]).weight_lattice(extended=True).fundamental_weights()[1]
        sage: GYW = crystals.infinity.GeneralizedYoungWalls(2)
        sage: y = GYW([[],[1,0],[2,1]])
        sage: y.in_highest_weight_crystal(La)
        True
        sage: x = GYW([[],[1],[2],[],[],[2],[],[],[2]])
        sage: x.in_highest_weight_crystal(La)
        False

    Python

        >>> from sage.all import *
        >>> La = RootSystem(['A',Integer(2),Integer(1)]).weight_lattice(extended=True).fundamental_weights()[Integer(1)]
        >>> GYW = crystals.infinity.GeneralizedYoungWalls(Integer(2))
        >>> y = GYW([[],[Integer(1),Integer(0)],[Integer(2),Integer(1)]])
        >>> y.in_highest_weight_crystal(La)
        True
        >>> x = GYW([[],[Integer(1)],[Integer(2)],[],[],[Integer(2)],[],[],[Integer(2)]])
        >>> x.in_highest_weight_crystal(La)
        False

  latex_large()

    Generate LaTeX code for `self` but the output is larger.

    This requires TikZ.

    EXAMPLES:

    Sage

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.latex_large()
        '\\begin{tikzpicture}[baseline=5,scale=.45] \n \\foreach \\x [count=\\s from 0] in \n{{},{1,0,3,2},{2,1},{3,2,1,0,3,2},{},{},{2}} \n{\\foreach \\y [count=\\t from 0] in \\x {  \\node[font=\\scriptsize] at (-\\t,\\s) {$\\y$}; \n \\draw (-\\t+.5,\\s+.5) to (-\\t-.5,\\s+.5); \n \\draw (-\\t+.5,\\s-.5) to (-\\t-.5,\\s-.5); \n \\draw (-\\t-.5,\\s-.5) to (-\\t-.5,\\s+.5);  } \n \\draw[-,thick] (.5,\\s+1) to (.5,-.5) to (-\\s-1,-.5); } \n \\end{tikzpicture} \n'

    Python

        >>> from sage.all import *
        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.latex_large()
        '\\begin{tikzpicture}[baseline=5,scale=.45] \n \\foreach \\x [count=\\s from 0] in \n{{},{1,0,3,2},{2,1},{3,2,1,0,3,2},{},{},{2}} \n{\\foreach \\y [count=\\t from 0] in \\x {  \\node[font=\\scriptsize] at (-\\t,\\s) {$\\y$}; \n \\draw (-\\t+.5,\\s+.5) to (-\\t-.5,\\s+.5); \n \\draw (-\\t+.5,\\s-.5) to (-\\t-.5,\\s-.5); \n \\draw (-\\t-.5,\\s-.5) to (-\\t-.5,\\s+.5);  } \n \\draw[-,thick] (.5,\\s+1) to (.5,-.5) to (-\\s-1,-.5); } \n \\end{tikzpicture} \n'

  number_of_parts()

    Return the value of \(\mathscr{N}\) on `self`.

    In [KLRS2016], the statistic \(\mathscr{N}\) was defined on elements in \(\mathcal{Y}(\infty)\) which counts how many parts are in the corresponding Kostant partition. Specifically, the computation of \(\mathscr{N}(Y)\) is done using the following algorithm:

    - If \(Y\) has no rows whose right-most box is colored \(n\) and such that the length of this row is a multiple of \(n+1\), then \(\mathscr{N}(Y)\) is the total number of distinct rows in \(Y\), not counting multiplicity.

    - Otherwise, search \(Y\) for the longest row such that the right-most box is colored \(n\) and such that the total number of boxes in the row is \(k(n+1)\) for some \(k\ge 1\). Replace this row by \(n+1\) distinct rows of length \(k\), reordering all rows, if necessary, so that the result is a proper wall. (Note that the resulting wall may no longer be reduced.) Repeat the search and replace process for all other rows of the above form for each \(k' < k\). Then \(\mathscr{N}(Y)\) is the number of distinct rows, not counting multiplicity, in the wall resulting from this process.

    EXAMPLES:

    Sage

        sage: Y = crystals.infinity.GeneralizedYoungWalls(3)
        sage: y = Y([[0],[],[],[],[0],[],[],[],[0]])
        sage: y.number_of_parts()
        1

        sage: Y = crystals.infinity.GeneralizedYoungWalls(3)
        sage: y = Y([[0,3,2],[1,0],[],[],[0,3],[1,0],[],[],[0]])
        sage: y.number_of_parts()
        4

        sage: Y = crystals.infinity.GeneralizedYoungWalls(2)
        sage: y = Y([[0,2,1],[1,0],[2,1,0,2,1,0,2,1,0],[],[2,1,0,2,1,0]])
        sage: y.number_of_parts()
        8

    Python

        >>> from sage.all import *
        >>> Y = crystals.infinity.GeneralizedYoungWalls(Integer(3))
        >>> y = Y([[Integer(0)],[],[],[],[Integer(0)],[],[],[],[Integer(0)]])
        >>> y.number_of_parts()
        1

        >>> Y = crystals.infinity.GeneralizedYoungWalls(Integer(3))
        >>> y = Y([[Integer(0),Integer(3),Integer(2)],[Integer(1),Integer(0)],[],[],[Integer(0),Integer(3)],[Integer(1),Integer(0)],[],[],[Integer(0)]])
        >>> y.number_of_parts()
        4

        >>> Y = crystals.infinity.GeneralizedYoungWalls(Integer(2))
        >>> y = Y([[Integer(0),Integer(2),Integer(1)],[Integer(1),Integer(0)],[Integer(2),Integer(1),Integer(0),Integer(2),Integer(1),Integer(0),Integer(2),Integer(1),Integer(0)],[],[Integer(2),Integer(1),Integer(0),Integer(2),Integer(1),Integer(0)]])
        >>> y.number_of_parts()
        8

  phi(i)

    Return the value \(\varepsilon_i(Y) + \langle h_i, \mathrm{wt}(Y)\rangle\), where \(h_i\) is the \(i\)-th simple coroot and \(Y\) is `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(3)([[0],[1,0,3,2],[2,1],[3,2,1,0,3,2],[0],[],[2]])
        sage: y.phi(1)
        3
        sage: y.phi(2)
        -1

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(0)],[],[Integer(2)]])
        >>> y.phi(Integer(1))
        3
        >>> y.phi(Integer(2))
        -1

  pp()

    Pretty print `self`.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(2)([[0,2,1],[1,0,2,1,0],[],[0],[1,0,2],[],[],[1]])
        sage: y.pp()
                1|
                 |
                 |
            2|0|1|
                0|
                 |
        0|1|2|0|1|
            1|2|0|

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(2))([[Integer(0),Integer(2),Integer(1)],[Integer(1),Integer(0),Integer(2),Integer(1),Integer(0)],[],[Integer(0)],[Integer(1),Integer(0),Integer(2)],[],[],[Integer(1)]])
        >>> y.pp()
                1|
                 |
                 |
            2|0|1|
                0|
                 |
        0|1|2|0|1|
            1|2|0|

  raw_signature(i)

    Return the sequence from \(\{+,-\}\) obtained from all \(i\)-admissible slots and removable \(i\)-boxes without canceling any \((+,-)\)-pairs. The result also notes the row and column of the sign.

    EXAMPLES:

    Sage

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.raw_signature(2)
        [['-', 3, 6], ['-', 1, 4], ['-', 6, 1]]

    Python

        >>> from sage.all import *
        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.raw_signature(Integer(2))
        [['-', 3, 6], ['-', 1, 4], ['-', 6, 1]]

  signature(i)

    Return the \(i\)-signature of `self`.

    The signature is obtained by reading `self` in columns bottom to top starting from the left. Then add a \(-\) at every \(i\)-box which may be removed from `self` and still obtain a legal generalized Young wall, and add a \(+\) at each site for which an \(i\)-box may be added and still obtain a valid generalized Young wall. Then successively cancel any \((+,-)\)-pair to obtain a sequence of the form \(- \cdots -+ \cdots +\). This resulting sequence is the output.

    EXAMPLES:

    Sage

        sage: y = crystals.infinity.GeneralizedYoungWalls(2)([[0],[1,0],[2,1,0,2],[],[1]])
        sage: y.signature(1)
        ''

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.signature(2)
        '---'

    Python

        >>> from sage.all import *
        >>> y = crystals.infinity.GeneralizedYoungWalls(Integer(2))([[Integer(0)],[Integer(1),Integer(0)],[Integer(2),Integer(1),Integer(0),Integer(2)],[],[Integer(1)]])
        >>> y.signature(Integer(1))
        ''

        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.signature(Integer(2))
        '---'

  sum_of_weighted_row_lengths()

    Return the value of \(\mathscr{M}\) on `self`.

    Let \(\mathcal{Y}_0 \subset \mathcal{Y}(\infty)\) be the set of generalized Young walls which have no rows whose right-most box is colored \(n\). For \(Y \in \mathcal{Y}_0\),

    \[\mathscr{M}(Y) = \sum_{i=1}^n (i+1)M_i(Y),\]

    where \(M_i(Y)\) is the number of nonempty rows in \(Y\) whose right-most box is colored \(i-1\).

    EXAMPLES:

    Sage

        sage: Y = crystals.infinity.GeneralizedYoungWalls(2)
        sage: y = Y([[0,2,1,0,2],[1,0,2],[],[0,2],[1,0],[],[0],[1,0]])
        sage: y.sum_of_weighted_row_lengths()
        15

    Python

        >>> from sage.all import *
        >>> Y = crystals.infinity.GeneralizedYoungWalls(Integer(2))
        >>> y = Y([[Integer(0),Integer(2),Integer(1),Integer(0),Integer(2)],[Integer(1),Integer(0),Integer(2)],[],[Integer(0),Integer(2)],[Integer(1),Integer(0)],[],[Integer(0)],[Integer(1),Integer(0)]])
        >>> y.sum_of_weighted_row_lengths()
        15

  weight(root_lattice=False)

    Return the weight of `self`.

    INPUT:

    - `root_lattice` – boolean determining whether weight should appear in root lattice or not in extended affine weight lattice

    EXAMPLES:

    Sage

        sage: x = crystals.infinity.GeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
        sage: x.weight()
        2*Lambda[0] + Lambda[1] - 4*Lambda[2] + Lambda[3] - 2*delta
        sage: x.weight(root_lattice=True)
        -2*alpha[0] - 3*alpha[1] - 5*alpha[2] - 3*alpha[3]

    Python

        >>> from sage.all import *
        >>> x = crystals.infinity.GeneralizedYoungWalls(Integer(3))([[],[Integer(1),Integer(0),Integer(3),Integer(2)],[Integer(2),Integer(1)],[Integer(3),Integer(2),Integer(1),Integer(0),Integer(3),Integer(2)],[],[],[Integer(2)]])
        >>> x.weight()
        2*Lambda[0] + Lambda[1] - 4*Lambda[2] + Lambda[3] - 2*delta
        >>> x.weight(root_lattice=True)
        -2*alpha[0] - 3*alpha[1] - 5*alpha[2] - 3*alpha[3]

class sage.combinat.crystals.generalized_young_walls.InfinityCrystalOfGeneralizedYoungWalls(n, category)

  Bases: `UniqueRepresentation`, `Parent`

  The crystal \(\mathcal{Y}(\infty)\) of generalized Young walls of type \(A_n^{(1)}\) as defined in [KS2010].

  A generalized Young wall is a collection of boxes stacked on a fixed board, such that color of the box at the site located in the \(j\)-th row from the bottom and the \(i\)-th column from the right is \(j-1 \bmod n+1\). There are several growth conditions on elements in \(Y \in \mathcal{Y}(\infty)\):

  - Walls grow in rows from right to left. That is, for every box \(y\in Y\) that is not in the rightmost column, there must be a box immediately to the right of \(y\).

  - For all \(p>q\) such that \(p-q \equiv 0 \bmod n+1\), the \(p\)-th row has most as many boxes as the \(q\)-th row.

  - There does not exist a column in the wall such that if one \(i\)-colored box, for every \(i = 0,1,\ldots,n\), is removed from that column, then the result satisfies the above conditions.

  There is a crystal structure on \(\mathcal{Y}(\infty)\) defined as follows. Define maps

  \[e_i,\ f_i \colon \mathcal{Y}(\infty) \longrightarrow \mathcal{Y}(\infty) \sqcup \{0\}, \qquad \varepsilon_i,\ \varphi_i \colon \mathcal{Y}(\infty) \longrightarrow \ZZ, \qquad \mathrm{wt}\colon \mathcal{Y}(\infty) \longrightarrow \bigoplus_{i=0}^n \ZZ \Lambda_i \oplus \ZZ \delta,\]

  by

  \[\mathrm{wt}(Y) = -\sum_{i=0}^n m_i(Y) \alpha_i,\]

  where \(m_i(Y)\) is the number of \(i\)-boxes in \(Y\), \(\varepsilon_i(Y)\) is the number of \(-\) in the \(i\)-signature of \(Y\), and

  \[\varphi_i(Y) = \varepsilon_i(Y) + \langle h_i, \mathrm{wt}(Y) \rangle.\]

  See `GeneralizedYoungWall.e()`, `GeneralizedYoungWall.f()`, and `GeneralizedYoungWall.signature()` for more about \(e_i\), \(f_i\), and \(i\)-signatures.

  INPUT:

  - `n` – type \(A_n^{(1)}\)

  EXAMPLES:

  Sage

      sage: Yinf = crystals.infinity.GeneralizedYoungWalls(3)
      sage: y = Yinf([[0],[1,0,3,2],[],[3,2,1],[0],[1,0]])
      sage: y.pp()
          0|1|
            0|
        1|2|3|
             |
      2|3|0|1|
            0|
      sage: y.weight(root_lattice=True)
      -4*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
      sage: y.f(0)
      [[0], [1, 0, 3, 2], [], [3, 2, 1], [0], [1, 0], [], [], [0]]
      sage: y.e(0).pp()
          0|1|
             |
        1|2|3|
             |
      2|3|0|1|
            0|

  Python

      >>> from sage.all import *
      >>> Yinf = crystals.infinity.GeneralizedYoungWalls(Integer(3))
      >>> y = Yinf([[Integer(0)],[Integer(1),Integer(0),Integer(3),Integer(2)],[],[Integer(3),Integer(2),Integer(1)],[Integer(0)],[Integer(1),Integer(0)]])
      >>> y.pp()
          0|1|
            0|
        1|2|3|
             |
      2|3|0|1|
            0|
      >>> y.weight(root_lattice=True)
      -4*alpha[0] - 3*alpha[1] - 2*alpha[2] - 2*alpha[3]
      >>> y.f(Integer(0))
      [[0], [1, 0, 3, 2], [], [3, 2, 1], [0], [1, 0], [], [], [0]]
      >>> y.e(Integer(0)).pp()
          0|1|
             |
        1|2|3|
             |
      2|3|0|1|
            0|

  To display the crystal down to depth 3:

  Sage

      sage: S = Yinf.subcrystal(max_depth=3)
      sage: G = Yinf.digraph(subset=S) # long time
      sage: view(G) # not tested

  [Python]

      >>> from sage.all import *
      >>> S = Yinf.subcrystal(max_depth=Integer(3))
      >>> G = Yinf.digraph(subset=S) # long time
      >>> view(G) # not tested

  Element

    alias of `GeneralizedYoungWall`