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 [KS10] and designed to be a realization of the crystals \mathcal{B}(\infty) and \mathcal{B}(\lambda) in type A_n^{(1)}.

REFERENCES:

[KS10](1, 2, 3, 4) J.-A. Kim and D.-U. Shin. Generalized Young walls and crystal bases for quantum affine algebra of type A. Proc. Amer. Math. Soc. 138(11), pp. 3877–3889, 2010.
[KLRS]S.-J. Kang, K.-H. Lee, H. Ryu, and B. Salisbury. A combinatorial description of the affine Gindikin-Karpelevich formula of type A_n^{(1)}. Arxiv 1203.1640.
class sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWalls(n, La)

Bases: sage.combinat.crystals.generalized_young_walls.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 [KS10]. 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().fundamental_weights()[1]
sage: YLa = CrystalOfGeneralizedYoungWalls(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]
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 = InfinityCrystalOfGeneralizedYoungWalls(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 = CrystalOfLSPaths(['A',3,1],[1,0,0,0])
sage: C = LS.subcrystal(max_depth=4)
sage: G = LS.digraph(subset=C)
sage: P = LS.weight_lattice_realization()
sage: La = P.fundamental_weights()
sage: YW = CrystalOfGeneralizedYoungWalls(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

To display the crystal down to a specified depth:

sage: S = YLa.subset(max_depth=4)
sage: sorted(list(S))
[[], [[], [1]], [[], [1], [2]], [[], [1], [2], [3]], [[], [1, 0]], [[], [1, 0], [2]], [[], [1, 0], [2], [3]], [[], [1, 0], [2, 1]], [[], [1, 0, 3]], [[], [1, 0, 3], [2]], [[], [1, 0, 3, 2]]]
sage: G = YLa.digraph(subset=S)
sage: view(G, tightpage=True) # not tested
Element

alias of CrystalOfGeneralizedYoungWallsElement

subset(max_depth=4)

Return a subset of self up to max_depth.

EXAMPLES:

sage: Y = CrystalOfGeneralizedYoungWalls(2,RootSystem(['A',2,1]).weight_lattice().fundamental_weights()[0])
sage: S = Y.subset(max_depth=3)
sage: S
[[], [[0]], [[0, 2]], [[0], [1]], [[0, 2, 1]], [[0, 2], [1]]]
class sage.combinat.crystals.generalized_young_walls.CrystalOfGeneralizedYoungWallsElement(parent, data)

Bases: sage.combinat.crystals.generalized_young_walls.GeneralizedYoungWall

Element of the highest weight crystal of generalized Young walls.

e(i)

Compute the action of \widetilde{e}_i restricted to the highest weight crystal.

EXAMPLES:

sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()[1]
sage: hwy = CrystalOfGeneralizedYoungWalls(2,La)([[],[1,0],[2,1]])
sage: hwy.e(1)
[[], [1, 0], [2]]
sage: hwy.e(2)
sage: hwy.e(3)
f(i)

Compute the action of \widetilde{f}_i restricted to the highest weight crystal.

EXAMPLES:

sage: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()[1]
sage: GYW = InfinityCrystalOfGeneralizedYoungWalls(2)
sage: y = GYW([[],[1,0],[2,1]])
sage: y.f(1)
[[], [1, 0], [2, 1], [], [1]]
sage: hwy = CrystalOfGeneralizedYoungWalls(2,La)([[],[1,0],[2,1]])
sage: hwy.f(1)
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: La = RootSystem(['A',2,1]).weight_lattice().fundamental_weights()[1]
sage: hwy = CrystalOfGeneralizedYoungWalls(2,La)([[],[1,0],[2,1]])
sage: hwy.weight()
Lambda[0] - Lambda[1] + Lambda[2]
class sage.combinat.crystals.generalized_young_walls.GeneralizedYoungWall(parent, data)

Bases: sage.combinat.combinat.CombinatorialObject, sage.structure.element.Element

A generalized Young wall.

For more information, see InfinityCrystalOfGeneralizedYoungWalls.

EXAMPLES:

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

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

EXAMPLES:

sage: y = InfinityCrystalOfGeneralizedYoungWalls(3)([[0],[1,0,3,2],[2,1],[3,2,1,0,3,2],[0],[],[2]])
sage: y.Epsilon()
Lambda[0] + 3*Lambda[2]
Phi()

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

EXAMPLES:

sage: y = InfinityCrystalOfGeneralizedYoungWalls(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=InfinityCrystalOfGeneralizedYoungWalls(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]
a(i, k)

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

EXAMPLES:

sage: y = InfinityCrystalOfGeneralizedYoungWalls(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
column(k)

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

EXAMPLES:

sage: y = InfinityCrystalOfGeneralizedYoungWalls(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 = InfinityCrystalOfGeneralizedYoungWalls(5)([])
sage: hw.column(1)
[]
content()

Return total number of blocks in self.

EXAMPLES:

sage: y = InfinityCrystalOfGeneralizedYoungWalls(2)([[0],[1,0],[2,1,0,2],[],[1]])
sage: y.content()
8

sage: x = InfinityCrystalOfGeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
sage: x.content()
13
e(i)

Return the application of the Kashiwara raising operator \widetilde{e}_i on self.

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

EXAMPLES:

sage: x=InfinityCrystalOfGeneralizedYoungWalls(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)
epsilon(i)

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

EXAMPLES:

sage: y=InfinityCrystalOfGeneralizedYoungWalls(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
f(i)

Return the application of the Kashiwara lowering operator \widetilde{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 = InfinityCrystalOfGeneralizedYoungWalls(2)([])
sage: hw.f(1)
[[], [1]]
sage: _.f(2)
[[], [1], [2]]
sage: _.f(0)
[[], [1, 0], [2]]
sage: _.f(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: y = InfinityCrystalOfGeneralizedYoungWalls(2)([[0],[1,0],[2,1,0,2],[],[1]])
sage: y.generate_signature(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 [KS10], 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().fundamental_weights()[1]
sage: GYW = InfinityCrystalOfGeneralizedYoungWalls(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
latex_large()

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

EXAMPLES:

sage: x = InfinityCrystalOfGeneralizedYoungWalls(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'
number_of_parts()

Return the value of

System Message: WARNING/2 (\mathscr{N})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \mathscr l.30 $\mathscr {N}$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 204 bytes). Transcript written on math.log.
on self.

In [KLRS], the statistic

System Message: WARNING/2 (\mathscr{N})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \mathscr l.30 $\mathscr {N}$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 204 bytes). Transcript written on math.log.
was defined on elements in \mathcal{Y}(\infty) which counts how many parts are in the corresponding Kostant partition. Specifically, the computation of

System Message: WARNING/2 (\mathscr{N}(Y))

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \mathscr l.30 $\mathscr {N}(Y)$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 260 bytes). Transcript written on math.log.
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

    System Message: WARNING/2 (\mathscr{N}(Y))

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \mathscr l.30 $\mathscr {N}(Y)$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 260 bytes). Transcript written on math.log.
    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

    System Message: WARNING/2 (\mathscr{N}(Y))

    latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \mathscr l.30 $\mathscr {N}(Y)$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 260 bytes). Transcript written on math.log.
    is the number of distinct rows, not counting multipicity, in the wall resulting from this process.

EXAMPLES:

sage: Y = InfinityCrystalOfGeneralizedYoungWalls(3)
sage: y = Y([[0],[],[],[],[0],[],[],[],[0]])
sage: y.number_of_parts()
1

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

sage: Y = InfinityCrystalOfGeneralizedYoungWalls(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
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: y = InfinityCrystalOfGeneralizedYoungWalls(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
pp()

Return an ASCII drawing of self.

EXAMPLES:

sage: y = InfinityCrystalOfGeneralizedYoungWalls(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|
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: x = InfinityCrystalOfGeneralizedYoungWalls(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]]
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: y = InfinityCrystalOfGeneralizedYoungWalls(2)([[0],[1,0],[2,1,0,2],[],[1]])
sage: y.signature(1)
''

sage: x = InfinityCrystalOfGeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
sage: x.signature(2)
'---'
sum_of_weighted_row_lengths()

Return the value of

System Message: WARNING/2 (\mathscr{M})

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <recently read> \mathscr l.30 $\mathscr {M}$ [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 204 bytes). Transcript written on math.log.
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,

System Message: WARNING/2 (\mathscr{M}(Y) = \sum_{i=1}^n (i+1)M_i(Y),)

latex exited with error: [stderr] [stdout] This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013) restricted \write18 enabled. entering extended mode (./math.tex LaTeX2e <2011/06/27> Babel <3.9f> and hyphenation patterns for 2 languages loaded. (/usr/share/texlive/texmf-dist/tex/latex/base/article.cls Document Class: article 2007/10/19 v1.4h Standard LaTeX document class (/usr/share/texlive/texmf-dist/tex/latex/base/size12.clo)) (/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/utf8x.def)) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucs.sty (/usr/share/texlive/texmf-dist/tex/latex/ucs/data/uni-global.def)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty For additional information on amsmath, use the `?’ option. (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty)) (/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty)) (/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty) (./math.aux) (/usr/share/texlive/texmf-dist/tex/latex/ucs/ucsencs.def) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd) (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd) ! Undefined control sequence. <argument> \begin {split}\mathscr {M}(Y) = \sum _{i=1}^n (i+1)M_i(Y),\end {... l.32 \end{gather} ! Undefined control sequence. <argument> \begin {split}\mathscr {M}(Y) = \sum _{i=1}^n (i+1)M_i(Y),\end {... l.32 \end{gather} [1] (./math.aux) ) (see the transcript file for additional information) Output written on math.dvi (1 page, 500 bytes). Transcript written on math.log.

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

EXAMPLES:

sage: Y = InfinityCrystalOfGeneralizedYoungWalls(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
weight()

Returns the weight of self as an element of the root lattice \bigoplus_{i=0}^n \ZZ \alpha_i.

EXAMPLES:

sage: x=InfinityCrystalOfGeneralizedYoungWalls(3)([[],[1,0,3,2],[2,1],[3,2,1,0,3,2],[],[],[2]])
sage: x.weight()
-2*alpha[0] - 3*alpha[1] - 5*alpha[2] - 3*alpha[3]
class sage.combinat.crystals.generalized_young_walls.InfinityCrystalOfGeneralizedYoungWalls(n, category)

Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation

The crystal \mathcal{Y}(\infty) of generalized Young walls of type A_n^{(1)} as defined in [KS10].

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

\widetilde{e}_i,\ \widetilde{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,

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 \widetilde{e}_i, \widetilde{f}_i, and i-signatures.

INPUT:

  • n – type A_n^{(1)}

EXAMPLES:

sage: Yinf = InfinityCrystalOfGeneralizedYoungWalls(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()
-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|

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, tightpage=True) # not tested
Element

alias of GeneralizedYoungWall

subset(max_depth=4)

Construct the subcrystal of self trucated at depth max_depth.

EXAMPLES:

sage: Y = InfinityCrystalOfGeneralizedYoungWalls(2)
sage: S = Y.subset(max_depth=2)
sage: S
[[], [[], [1]], [[], [], [2]], [[0]], [[0, 2]], [[0], [1]], [[], [], [2], [], [], [2]],
[[], [1], [2]], [[0], [], [], [0]], [[0], [], [2]], [[], [], [2, 1]], [[], [1], [], [], [1]], [[], [1, 0]]]

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