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r""" `\mathcal{B}(\infty)` Crystals of Tableaux in Nonexceptional Types and `G_2`
A tableau model for `\mathcal{B}(\infty)`. For more information, see :class:`~sage.combinat.crystals.infinity_crystals.InfinityCrystalOfTableaux`.
AUTHORS:
- Ben Salisbury: Initial version
- Travis Scrimshaw: Initial version """
#***************************************************************************** # Copyright (C) 2013 Ben Salisbury <bsalisbury1 at gmail.com> # Travis Scrimshaw <tscrim at ucdavis.edu> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #****************************************************************************
from sage.structure.parent import Parent from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets from sage.categories.highest_weight_crystals import HighestWeightCrystals from sage.categories.homset import Hom from sage.misc.cachefunc import cached_method from sage.misc.flatten import flatten
from sage.combinat.partition import Partition from sage.combinat.root_system.cartan_type import CartanType from sage.combinat.crystals.letters import CrystalOfLetters from sage.combinat.crystals.tensor_product import CrystalOfWords from sage.combinat.crystals.tensor_product_element import (CrystalOfTableauxElement, InfinityCrystalOfTableauxElement, InfinityCrystalOfTableauxElementTypeD)
class InfinityCrystalOfTableaux(CrystalOfWords): r""" `\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.
.. rubric:: 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 \}`.
.. rubric:: 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.
.. rubric:: 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}`.
.. rubric:: 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.
.. rubric:: 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] \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 = 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 """ @staticmethod def __classcall_private__(cls, cartan_type): """ Normalize input to ensure a unique representation.
EXAMPLES::
sage: B = crystals.infinity.Tableaux(['A',4]) sage: B2 = crystals.infinity.Tableaux(CartanType(['A',4])) sage: B is B2 True """
def __init__(self, cartan_type): """ Initialize ``self``.
EXAMPLES::
sage: B = crystals.infinity.Tableaux(['A',2]) sage: TestSuite(B).run() # long time """ InfiniteEnumeratedSets()))
def _repr_(self): """ Return a string representation of ``self``.
EXAMPLES::
sage: B = crystals.infinity.Tableaux(['A',4]); B The infinity crystal of tableaux of type ['A', 4] """
@cached_method def module_generator(self): """ 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 = crystals.infinity.Tableaux(['A',3]) sage: T.module_generator() [[1, 1, 1], [2, 2], [3]] """ # The column canonical tableau, read by columns
def _element_constructor_(self, *args, **options): """ Construct an element of ``self`` from the input data.
EXAMPLES::
sage: T = crystals.Tableaux(['A',3], shape = [2,2]) sage: T(rows=[[1,2],[3,4]]) [[1, 2], [3, 4]] sage: T(columns=[[3,1],[4,2]]) [[1, 2], [3, 4]] """
def _coerce_map_from_(self, P): """ Return ``True`` or the coerce map from ``P`` if a map exists.
EXAMPLES::
sage: T = crystals.infinity.Tableaux(['A',3]) sage: RC = crystals.infinity.RiggedConfigurations(['A',3]) sage: T._coerce_map_from_(RC) Crystal Isomorphism morphism: From: The infinity crystal of rigged configurations of type ['A', 3] To: The infinity crystal of tableaux of type ['A', 3] """ InfinityCrystalOfNonSimplyLacedRC) and (self.cartan_type().is_simply_laced() or isinstance(P, InfinityCrystalOfNonSimplyLacedRC))):
class Element(InfinityCrystalOfTableauxElement): r""" Elements in `\mathcal{B}(\infty)` crystal of tableaux. """ def phi(self,i): r""" 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 :meth:`weight` of `T`.
INPUT:
- ``i`` -- An element of the index set
EXAMPLES::
sage: B = crystals.infinity.Tableaux("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 = crystals.infinity.Tableaux("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] """
@cached_method def weight(self): r""" 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
.. MATH::
\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 = crystals.infinity.Tableaux("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 = crystals.infinity.Tableaux("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 = crystals.infinity.Tableaux("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 """ (cur_col_len == n and ty == 'B'): else: # Since we miss the last column (which is always height 1)
def reduced_form(self): r""" 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 = crystals.infinity.Tableaux(['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() [['*'], [4, 4, 4], [4, 4]] """ else:
def seg(self): r""" 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 = crystals.infinity.Tableaux(['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 = crystals.infinity.Tableaux(['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 = crystals.infinity.Tableaux(['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 """ for r in range(len(tab)): for c in range(len(tab[r])): if tab[r][c] == 0 and tab[r][-1] == -r-1: segments.remove([r,tab[r][c]]) segments.remove([0,tab[0][c]])
def content(self): r""" 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 = crystals.infinity.Tableaux("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 = crystals.infinity.Tableaux("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 = crystals.infinity.Tableaux("C2") sage: b = B(rows=[[1,1,1,1,1,1,2,2,2,-2,-2],[2,-2,-2,-2]]) sage: b.content() 8 """
class InfinityCrystalOfTableauxTypeD(InfinityCrystalOfTableaux): r""" `\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 :class:`~sage.combinat.crystals.infinity_crystals.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) """ @staticmethod def __classcall_private__(cls, cartan_type): """ Normalize input to ensure a unique representation.
EXAMPLES::
sage: B = crystals.infinity.Tableaux(['D',4]) sage: B2 = crystals.infinity.Tableaux(CartanType(['D',4])) sage: B is B2 True """
@cached_method def module_generator(self): """ 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 = crystals.infinity.Tableaux(['D',4]) sage: T.module_generator() [[1, 1, 1], [2, 2], [3]] """ # The column canonical tableau, read by columns
class Element(InfinityCrystalOfTableauxElementTypeD, InfinityCrystalOfTableaux.Element): r""" Elements in `\mathcal{B}(\infty)` crystal of tableaux for type `D_n`. """ pass
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