\(\mathcal{B}(\infty)\) Crystals of Tableaux in Nonexceptional Types and \(G_2\)¶
A tableau model for \(\mathcal{B}(\infty)\). For more information, see
InfinityCrystalOfTableaux
.
AUTHORS:
- Ben Salisbury: Initial version
- Travis Scrimshaw: Initial version
-
sage.combinat.crystals.infinity_crystals.
InfinityCrystalOfTableaux
¶ \(\mathcal{B}(\infty)\) crystal of tableaux.
A tableaux model \(\mathcal{T}(\infty)\) for the crystal \(\mathcal{B}(\infty)\) is introduced by Hong and Lee in [HL2008]. This model is currently valid for types \(A_n\), \(B_n\), \(C_n\), \(D_n\), and \(G_2\), and builds on the tableaux model given by Kashiwara and Nakashima [KN1994] in types \(A_n\), \(B_n\), \(C_n\), and \(D_n\), and by Kang and Misra [KM1994] in type \(G_2\).
Note
We are using the English convention for our tableaux.
We say a tableau \(T\) is marginally large if:
- for each \(1 \leq i \leq n\), the leftmost box in the \(i\)-th row from the top in \(T\) is an \(i\)-box,
- for each \(1 \leq i \leq n\), the number of \(i\)-boxes in the \(i\)-th row from the top in \(T\) is greater than the total number of boxes in the \((i+1)\)-th row by exactly one.
We now will describe this tableaux model type-by-type.
Type \(A_n\)
\(\mathcal{T}(\infty)\) is the set of marginally large semistandard tableaux with exactly \(n\) rows over the alphabet \(\{1 \prec 2 \prec \cdots \prec n+1 \}\).
Type \(B_n\)
\(\mathcal{T}(\infty)\) is the set of marginally large semistandard tableaux with exactly \(n\) rows over the alphabet \(\{1 \prec \cdots \prec n \prec 0 \prec \overline{n} \prec \cdots \prec \overline{1} \}\) and subject to the following constraints:
- for each \(1 \le i \le n\), the contents of the boxes in the \(i\)-th row are \(\preceq \overline{i}\),
- the entry \(0\) can appear at most once in a single row.
Type \(C_n\)
\(\mathcal{T}(\infty)\) is the set of marginally large semistandard tableaux with exactly \(n\) rows over the alphabet \(\{1 \prec \cdots \prec n \prec \overline{n} \prec \cdots \prec \overline{1} \}\) and for each \(1 \leq i \leq n\), the contents of the boxes in the \(i\)-th row are \(\preceq \overline{i}\).
Type \(D_n\)
\(\mathcal{T}(\infty)\) is the set of marginally large semistandard tableaux with exactly \(n-1\) rows over the alphabet \(\{1 \prec \cdots \prec n, \overline{n} \prec \cdots \prec \overline{1} \}\) and subject to the following constraints:
- for each \(1 \le i \le n\), the contents of the boxes in the \(i\)-th row are \(\preceq \overline{i}\),
- the entries \(n\) and \(\overline{n}\) may not appear simultaneously in a single row.
Type \(G_2\)
\(\mathcal{T}(\infty)\) is the set of marginally large semistandard tableaux with exactly \(2\) rows over the ordered alphabet \(\{1 \prec 2 \prec 3 \prec 0 \prec \overline{3} \prec \overline{2} \prec \overline{1}\}\) and subject to the following constraints:
- the contents of the boxes in the first row are \(\preceq \overline{i}\),
- the contents of the boxes in the second row are \(\preceq 3\),
- the entry \(0\) can appear at most once in the first row and not at all in the second row.
In particular, the shape of the tableaux is not fixed in any instance of \(\mathcal{T}(\infty)\); the row lengths of a tableau can be arbitrarily long.
INPUT:
cartan_type
– One of['A',n]
,['B',n]
,['C',n]
,['D',n]
, or['G',2]
, wheren
is a positive integer
EXAMPLES:
sage: B = crystals.infinity.Tableaux(['A',2]) sage: b = B.highest_weight_vector(); b.pp() 1 1 2 sage: b.f_string([2,1,1,2,2,2]).pp() 1 1 1 1 1 2 3 2 3 3 3 sage: B = crystals.infinity.Tableaux(['G',2]) sage: b = B(rows=[[1,1,1,1,1,2,3,3,0,-3,-1,-1,-1],[2,3,3,3]]) sage: b.e_string([2,1,1,1,1,1,1]).pp() 1 1 1 1 2 3 3 3 3 -2 -2 -2 2 3 3 sage: b.e_string([2,1,1,1,1,1,1,1])
We check that a few classical crystals embed into \(\mathcal{T}(\infty)\):
sage: def crystal_test(B, C): ....: T = crystals.elementary.T(C.cartan_type(), C.module_generators[0].weight()) ....: TP = crystals.TensorProduct(T, B) ....: mg = TP(T[0], B.module_generators[0]) ....: g = {C.module_generators[0]: mg} ....: f = C.crystal_morphism(g, category=HighestWeightCrystals()) ....: G = B.digraph(subset=[f(x) for x in C]) ....: return G.is_isomorphic(C.digraph(), edge_labels=True) sage: B = crystals.infinity.Tableaux(['A',2]) sage: C = crystals.Tableaux(['A',2], shape=[2,1]) sage: crystal_test(B, C) True sage: C = crystals.Tableaux(['A',2], shape=[6,2]) sage: crystal_test(B, C) True sage: B = crystals.infinity.Tableaux(['B',2]) sage: C = crystals.Tableaux(['B',2], shape=[3]) sage: crystal_test(B, C) True sage: C = crystals.Tableaux(['B',2], shape=[2,1]) sage: crystal_test(B, C) True sage: B = crystals.infinity.Tableaux(['C',3]) sage: C = crystals.Tableaux(['C',3], shape=[2,1]) sage: crystal_test(B, C) True sage: B = crystals.infinity.Tableaux(['D',4]) sage: C = crystals.Tableaux(['D',4], shape=[2]) sage: crystal_test(B, C) True sage: C = crystals.Tableaux(['D',4], shape=[1,1,1,1]) sage: crystal_test(B, C) True sage: B = crystals.infinity.Tableaux(['G',2]) sage: C = crystals.Tableaux(['G',2], shape=[3]) sage: crystal_test(B, C) True
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sage.combinat.crystals.infinity_crystals.
InfinityCrystalOfTableauxTypeD
¶ \(\mathcal{B}(\infty)\) crystal of tableaux for type \(D_n\).
This is the set \(\mathcal{T}(\infty)\) of marginally large semistandard tableaux with exactly \(n-1\) rows over the alphabet \(\{1 \prec \cdots \prec n, \overline{n} \prec \cdots \prec \overline{1} \}\) and subject to the following constraints:
- for each \(1 \le i \le n\), the contents of the boxes in the \(i\)-th row are \(\preceq \overline{i}\),
- the entries \(n\) and \(\overline{n}\) may not appear simultaneously in a single row.
For more information, see
InfinityCrystalOfTableaux
.EXAMPLES:
sage: B = crystals.infinity.Tableaux("D4") sage: b = B.highest_weight_vector().f_string([4,3,2,1,4]) sage: b.pp() 1 1 1 1 1 1 2 2 2 2 2 3 3 -4 -3 sage: b.weight() (-1, 0, -2, -1)