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 = 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]
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 = LS.weight_lattice_realization()
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

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 = crystals.GeneralizedYoungWalls(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 = crystals.GeneralizedYoungWalls(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 = 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)

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 = crystals.GeneralizedYoungWalls(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 = 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|
   |

Epsilon()

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]

Phi()

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]

a(i, k)

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

column(k)

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)
[]

content()

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

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 = 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)

epsilon(i)

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

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 = 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]]

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 = crystals.infinity.GeneralizedYoungWalls(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 = 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

latex_large()

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'

number_of_parts()

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

In [KLRS], 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 multipicity, 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

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 = 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

pp()

Return an ASCII drawing of 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|

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 = 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]]

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 = 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)
'---'

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: 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

weight()

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

EXAMPLES:

sage: x = crystals.infinity.GeneralizedYoungWalls(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 = 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()
-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 = crystals.infinity.GeneralizedYoungWalls(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]]]