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:
-
sage.combinat.crystals.generalized_young_walls.
CrystalOfGeneralizedYoungWalls
¶ 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
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
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sage.combinat.crystals.generalized_young_walls.
CrystalOfGeneralizedYoungWallsElement
¶ Element of the highest weight crystal of generalized Young walls.
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sage.combinat.crystals.generalized_young_walls.
GeneralizedYoungWall
¶ 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| |
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sage.combinat.crystals.generalized_young_walls.
InfinityCrystalOfGeneralizedYoungWalls
¶ 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()
, andGeneralizedYoungWall.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|
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