Crystals of Generalized Young Walls#

AUTHORS:

  • Lucas David-Roesler: Initial version

  • Ben Salisbury: Initial version

  • Travis Scrimshaw: Initial version

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:

class sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWalls(n, La)[source]#

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: 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

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

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

alias of CrystalOfGeneralizedYoungWallsElement

class sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWallsElement(parent, data)[source]#

Bases: GeneralizedYoungWall

Element of the highest weight crystal of generalized Young walls.

e(i)[source]#

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

EXAMPLES:

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)

>>> 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)[source]#

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

EXAMPLES:

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)

>>> 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)[source]#

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: 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

>>> 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()[source]#

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: 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

>>> 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)[source]#

Bases: CombinatorialElement

A generalized Young wall.

For more information, see InfinityCrystalOfGeneralizedYoungWalls.

EXAMPLES:

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|
   |

>>> 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()[source]#

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

EXAMPLES:

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]

>>> 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()[source]#

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

EXAMPLES:

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]

>>> 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)[source]#

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

EXAMPLES:

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

>>> 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)[source]#

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

EXAMPLES:

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)
[]

>>> 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()[source]#

Return total number of blocks in self.

EXAMPLES:

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

>>> 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)[source]#

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: 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)

>>> 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)[source]#

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

EXAMPLES:

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

>>> 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)[source]#

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: 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]]

>>> 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)[source]#

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

>>> 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)[source]#

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: 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

>>> 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()[source]#

Generate LaTeX code for self but the output is larger. Requires TikZ.

EXAMPLES:

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 (-\\t-1,-.5); } \n \\end{tikzpicture} \n'

>>> 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 (-\\t-1,-.5); } \n \\end{tikzpicture} \n'
number_of_parts()[source]#

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: 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

>>> 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)[source]#

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: 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

>>> 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()[source]#

Pretty print self.

EXAMPLES:

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|

>>> 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)[source]#

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: 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]]

>>> 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)[source]#

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: 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)
'---'

>>> 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()[source]#

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: 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

>>> 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)[source]#

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: 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]

>>> 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)[source]#

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: 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|

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

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

alias of GeneralizedYoungWall