\(\mathcal{B}(\infty)\) 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
class sage.combinat.crystals.infinity_crystals.InfinityCrystalOfTableaux(cartan_type)

Bases: sage.combinat.crystals.tensor_product.CrystalOfWords

\(\mathcal{B}(\infty)\) crystal of tableaux.

A tableaux model \(\mathcal{T}(\infty)\) for the crystal \(\mathcal{B}(\infty)\) is introduced by Hong and Lee in [HL08]. 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 [KN94] in types \(A_n\), \(B_n\), \(C_n\), and \(D_n\), and by Kang and Misra [KM94] 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 constaints:

  • 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.

REFERENCES:

[BN10]D. Bump and M. Nakasuji. Integration on \(p\)-adic groups and crystal bases. Proc. Amer. Math. Soc. 138(5), pp. 1595–1605.
[LS12](1, 2) K.-H. Lee and B. Salisbury. Young tableaux, canonical bases, and the Gindikin-Karpelevich formula. Arxiv 1205.6006.
[HL08]J. Hong and H. Lee. Young tableaux and crystal \(B(\infty)\) for finite simple Lie algebras. J. Algebra 320, pp. 3680–3693, 2008.
[KM94]S.-J. Kang and K. C. Misra. Crystal bases and tensor product decompositions of \(U_q(G_2)\)-modules. J. Algebra 163, pp. 675–691, 1994.

INPUT:

  • cartan_type – One of ['A',n], ['B',n], ['C',n], ['D',n], or ['G',2], where n is a positive integer

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['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 = InfinityCrystalOfTableaux(['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):
....:     g = {C.module_generators[0] : B.module_generators[0]}
....:     f = C.crystal_morphism(g)
....:     G = B.digraph(subset=[f(x) for x in C])
....:     return G.is_isomorphic(C.digraph(), edge_labels=True)
sage: B = InfinityCrystalOfTableaux(['A',2])
sage: C = CrystalOfTableaux(['A',2], shape=[2,1])
sage: crystal_test(B, C)
True
sage: C = CrystalOfTableaux(['A',2], shape=[6,2])
sage: crystal_test(B, C)
True
sage: B = InfinityCrystalOfTableaux(['B',2])
sage: C = CrystalOfTableaux(['B',2], shape=[3])
sage: crystal_test(B, C)
True
sage: C = CrystalOfTableaux(['B',2], shape=[2,1])
sage: crystal_test(B, C)
True
sage: B = InfinityCrystalOfTableaux(['C',3])
sage: C = CrystalOfTableaux(['C',3], shape=[2,1])
sage: crystal_test(B, C)
True
sage: B = InfinityCrystalOfTableaux(['D',4])
sage: C = CrystalOfTableaux(['D',4], shape=[2])
sage: crystal_test(B, C)
True
sage: C = CrystalOfTableaux(['D',4], shape=[1,1,1,1])
sage: crystal_test(B, C)
True
sage: B = InfinityCrystalOfTableaux(['G',2])
sage: C = CrystalOfTableaux(['G',2], shape=[3])
sage: crystal_test(B, C)
True
class Element(parent, *args, **options)

Bases: sage.combinat.crystals.tensor_product.CrystalOfTableauxElement

Elements in \(\mathcal{B}(\infty)\) crystal of tableaux.

content()

Return the content of self.

The content \(|T|\) of \(T \in \mathcal{B}(\infty)\) is the number of blocks added to the highest weight to obtain \(T\) with any \(\overline{\imath}\)-boxes in the \(i\)-th row counted with multiplicity \(2\) provided the underlying Cartan type is of type \(B\), \(D\), or \(G\).

EXAMPLES:

sage: B = InfinityCrystalOfTableaux("D5")
sage: b = B.highest_weight_vector().f_string([5,4,3,1,1,3,4,5,3,4,5,1,4,5,2,3,5,3,2,4])
sage: b.content()
13

sage: B = InfinityCrystalOfTableaux("B2")
sage: b = B(rows=[[1,1,1,1,1,1,2,2,2,-2,-2],[2,0,-2,-2,-2]])
sage: b.content()
12

sage: B = InfinityCrystalOfTableaux("C2")
sage: b = B(rows=[[1,1,1,1,1,1,2,2,2,-2,-2],[2,-2,-2,-2]])
sage: b.content()
8
e(i)

Return the action of \(\widetilde{e}_i\) on self.

INPUT:

  • i – An element of the index set

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['B',3])
sage: b = B(rows=[[1,1,1,1,1,1,1,2,0,-3,-1,-1,-1,-1],[2,2,2,2,-2,-2],[3,-3,-3]])
sage: b.e(3).pp()
1  1  1  1  1  1  1  2  0 -3 -1 -1 -1 -1
2  2  2  2 -2 -2
3  0 -3
sage: b.e(1).pp()
1  1  1  1  1  1  1  0 -3 -1 -1 -1 -1
2  2  2  2 -2 -2
3 -3 -3
f(i)

Return the action of \(\widetilde{f}_i\) on self.

INPUT:

  • i – An element of the index set

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['C',4])
sage: b = B.highest_weight_vector()
sage: b.f(1).pp()
1  1  1  1  2
2  2  2
3  3
4
sage: b.f(3).pp()
1  1  1  1  1
2  2  2  2
3  3  4
4
sage: b.f(3).f(4).pp()
1  1  1  1  1
2  2  2  2
3  3 -4
4
phi(i)

Return \(\varphi_i\) of self.

Let \(T \in \mathcal{B}(\infty)\) Define \(\varphi_i(T) := \varepsilon_i(T) + \langle h_i, \mathrm{wt}(T) \rangle\), where \(h_i\) is the \(i\)-th simple coroot and \(\mathrm{wt}(T)\) is the weight() of \(T\).

INPUT:

  • i – An element of the index set

EXAMPLES:

sage: B = InfinityCrystalOfTableaux("A3")
sage: [B.highest_weight_vector().f_string([1,3,2,3,1,3,2,1]).phi(i) for i in B.index_set()]
[-3, 4, -3]

sage: B = InfinityCrystalOfTableaux("G2")
sage: [B.highest_weight_vector().f_string([2,2,1,2,1,1,1,2]).phi(i) for i in B.index_set()]
[5, -3]
reduced_form()

Return the reduced form of self.

The reduced form of a tableaux \(T \in \mathcal{T}(\infty)\) is the (not necessarily semistandard) tableaux obtained from \(T\) by removing all \(i\)-boxes in the \(i\)-th row, subject to the condition that if the row is empty, a \(\ast\) is put as a placeholder. This is described in [BN10] and [LS12].

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['A',3])
sage: b = B.highest_weight_vector().f_string([2,2,2,3,3,3,3,3])
sage: b.pp()
1  1  1  1  1  1  1  1
2  2  2  2  4  4  4
3  4  4
sage: b.reduced_form().pp()
*
4  4  4
4  4
seg()

Returns the statistic \(\mathrm{seg}\) of self.

More precisely, following [LS12], define a \(k\)-segment of a tableau \(T\) in \(\mathcal{B}(\infty)\) to be a maximal string of \(k\)-boxes in a single row of \(T\). Set \(\mathrm{seg}^{\prime}(T)\) to be the number of \(k\)-segments in \(T\), as \(k\) varies over all possible values. Then \(\mathrm{seg}(T)\) is determined type-by-type.

  • In types \(A_n\) and \(C_n\), define \(\mathrm{seg}(T) := \mathrm{seg}^{\prime}(T)\).
  • In types \(B_n\) and \(G_2\), set \(e(T)\) to be the number of rows in \(T\) which contain both a \(0\)-box and an \(\overline{\imath}\)-box. Define \(\mathrm{seg}(T) := \mathrm{seg}^{\prime}(T) - e(T)\).
  • In type \(D_n\), set \(d(T)\) to be the number of rows in \(T\) which contain an \(\overline{\imath}\)-box, but no \(n\)-box nor \(\overline{n}\)-box. Define \(\mathrm{seg}(T) := \mathrm{seg}^{\prime}(T) + d(T)\).

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['A',3])
sage: b = B.highest_weight_vector().f_string([1,3,2,2,3,1,1,3])
sage: b.pp()
1  1  1  1  1  1  2  2  4
2  2  2  2  3
3  4  4
sage: b.seg()
4

sage: B = InfinityCrystalOfTableaux(['D',4])
sage: b = B(rows=[[1,1,1,1,1,1,3,-2,-1],[2,2,2,4,-2],[3,3],[4]])
sage: b.pp()
1  1  1  1  1  1  3 -2 -1
2  2  2  4 -2
3  3
4
sage: b.seg()
6

sage: B = InfinityCrystalOfTableaux(['G',2])
sage: b = B.highest_weight_vector().f_string([2,1,1,1,2,1,2,2,1,2,2,2,1,2,2,1])
sage: b.pp()
1  1  1  1  1  1  1  1  2  3  0 -3
2  3  3  3  3  3  3
sage: b.seg()
5
string_parameters(word)

Return the string parametrization of self with respect to word.

For \(w\) in the Weyl group, let \(R(w)\) denote the set of reduced expressions for \(w\); that is, if \(w = s_{i_1} s_{i_2} \cdots s_{i_{\ell}}\) is a reduced expression, then \((i_1, i_2, \ldots, i_{\ell}) \in R(w)\). For brevity, such words are denoted by \(\mathbf{i}\).

For \(T \in \mathcal{T}(\infty)\) and \(\mathbf{i} = (i_1, \ldots, i_{\ell}) \in R(w)\), the string parametrization \((a_1, \ldots, a_{\ell})\) of \(T\) in the direction \(\mathbf{i}\) is defined recursively by \(a_1 = \varepsilon_{i_1}(T)\) and \(a_j = \varepsilon_{i_j} (\widetilde{e}_{i_{j-1}}^{\, a_{j-1}} \cdots \widetilde{e}_{i_1}^{\, a_1} T)\) for \(j = 2, \ldots, \ell\). If \(w = w_0\) is the longest element of the Weyl group, then the path determined by the string parametrization terminates at the highest weight vector.

INPUT:

  • word – A word in the alphabet of the index set

EXAMPLES:

sage: B = InfinityCrystalOfTableaux("A5")
sage: b = B(rows=[[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,6,6,6,6,6,6],[2,2,2,2,2,2,2,2,2,4,5,5,5,6],[3,3,3,3,3,3,3,5],[4,4,4,6,6,6],[5,6]])
sage: b.string_parameters([1,2,1,3,2,1,4,3,2,1,5,4,3,2,1])
[0, 1, 1, 1, 1, 0, 4, 4, 3, 0, 11, 10, 7, 7, 6]

sage: B = InfinityCrystalOfTableaux("G2")
sage: b = B(rows=[[1,1,1,1,1,3,3,0,-3,-3,-2,-2,-1,-1,-1,-1],[2,3,3,3]])
sage: b.string_parameters([2,1,2,1,2,1])
[5, 13, 11, 15, 4, 4]
sage: b.string_parameters([1,2,1,2,1,2])
[7, 12, 15, 8, 10, 0]
weight()

Return the weight of self.

From the definition of a crystal and that the highest weight element \(b_{\infty}\) of \(\mathcal{B}(\infty)\) is \(0\), the weight of \(T \in \mathcal{B}(\infty)\) can be defined as \(\mathrm{wt}(T) := -\sum_j \alpha_{i_j}\) where \(\widetilde{e}_{i_1} \cdots \widetilde{e}_{i_{\ell}} T = b_{\infty}\) and \(\{\alpha_i\}\) is the set of simple roots. (Note that the weight is independent of the path chosen to get to the highest weight.)

However we can also take advantage of the fact that \(\rho \colon R_{\lambda} \otimes \mathcal{B}(\infty) \longrightarrow B(\lambda)\), where \(\lambda\) is the shape of \(T\), preserves the tableau representation of \(T\). Therefore

\[\mathrm{wt}(T) = \mathrm{wt}\bigl( \rho(T) \bigr) - \lambda\]

where \(\mathrm{wt}\bigl( \rho(T) \bigr)\) is just the usual weight of the tableau \(T\).

Let \(\Lambda_i\) be the \(i\)-th fundamental weight. In type \(D\), the height \(n-1\) columns corresponds to \(\Lambda_{n-1} + \Lambda_n\) and the in type \(B\), the height \(n\) columns corresponds to \(2 \Lambda_n\).

EXAMPLES:

sage: B = InfinityCrystalOfTableaux("C7")
sage: b = B.highest_weight_vector().f_string([1,6,4,7,4,2,4,6,2,4,6,7,1,2,4,7])
sage: b.weight()
(-2, -1, 3, -5, 5, -3, -3)

Check that the definitions agree:

sage: P = B.weight_lattice_realization()
sage: alpha = P.simple_roots()
sage: b.weight() == -2*alpha[1] - 3*alpha[2] - 5*alpha[4] - 3*alpha[6] - 3*alpha[7]
True

Check that it works for type \(B\):

sage: B = InfinityCrystalOfTableaux("B2")
sage: B.highest_weight_vector().weight()
(0, 0)
sage: b = B.highest_weight_vector().f_string([1,2,2,2,1,2])
sage: P = B.weight_lattice_realization()
sage: alpha = P.simple_roots()
sage: b.weight() == -2*alpha[1] - 4*alpha[2]
True

Check that it works for type \(D\):

sage: B = InfinityCrystalOfTableaux("D4")
sage: B.highest_weight_vector().weight()
(0, 0, 0, 0)
sage: b = B.highest_weight_vector().f_string([1,4,4,2,4,3,2,4,1,3,2,4])
sage: P = B.weight_lattice_realization()
sage: alpha = P.simple_roots()
sage: b.weight() == -2*alpha[1] - 3*alpha[2] - 2*alpha[3] - 5*alpha[4]
True
InfinityCrystalOfTableaux.module_generator()

Return the module generator (or highest weight element) of self.

The module generator is the unique tableau of shape \((n, n-1, \ldots, 2, 1)\) with weight \(0\).

EXAMPLES:

sage: T = InfinityCrystalOfTableaux(['A',3])
sage: T.module_generator()
[[1, 1, 1], [2, 2], [3]]
class sage.combinat.crystals.infinity_crystals.InfinityCrystalOfTableauxTypeD(cartan_type)

Bases: sage.combinat.crystals.infinity_crystals.InfinityCrystalOfTableaux

\(\mathcal{B}(\infty)\) 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 = InfinityCrystalOfTableaux("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)
class Element(parent, *args, **options)

Bases: sage.combinat.crystals.infinity_crystals.InfinityCrystalOfTableaux.Element

Elements in \(\mathcal{B}(\infty)\) crystal of tableaux for type \(D_n\).

e(i)

Return the action of \(\widetilde{e}_i\) on self.

INPUT:

  • i – An element of the index set

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['D',4])
sage: b = B.highest_weight_vector().f_string([1,4,3,1,2]); b.pp()
1  1  1  1  2  3
2  2  2
3 -3
sage: b.e(2).pp()
1  1  1  1  2  2
2  2  2
3 -3
f(i)

Return the action of \(\widetilde{f}_i\) on self.

INPUT:

  • i – An element of the index set

EXAMPLES:

sage: B = InfinityCrystalOfTableaux(['D',5])
sage: b = B.highest_weight_vector().f_string([1,4,3,1,5]); b.pp()
1  1  1  1  1  1  2  2
2  2  2  2  2
3  3  3 -5
4  5
sage: b.f(1).pp()
1  1  1  1  1  1  2  2  2
2  2  2  2  2
3  3  3 -5
4  5
sage: b.f(5).pp()
1  1  1  1  1  1  2  2
2  2  2  2  2
3  3  3 -5
4 -4
InfinityCrystalOfTableauxTypeD.module_generator()

Return the module generator (or highest weight element) of self.

The module generator is the unique tableau of shape \((n-1, \ldots, 2, 1)\) with weight \(0\).

EXAMPLES:

sage: T = InfinityCrystalOfTableaux(['D',4])
sage: T.module_generator()
[[1, 1, 1], [2, 2], [3]]

Previous topic

Crystals of Generalized Young Walls

Next topic

Crystals of Modified Nakajima Monomials

This Page