summaryrefslogtreecommitdiffstats
path: root/src/sage/combinat/crystals/subcrystal.py
blob: c1bfd52854f279ad7a0b377c2cb3252d995a8fe0 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
r"""
Subcrystals

These are the crystals that are subsets of a larger ambient crystal.

AUTHORS:

- Travis Scrimshaw (2013-10-16): Initial implementation
"""

#*****************************************************************************
#       Copyright (C) 2013 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.misc.lazy_attribute import lazy_attribute
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.element import parent
from sage.structure.parent import Parent
from sage.structure.element_wrapper import ElementWrapper
from sage.categories.crystals import Crystals
from sage.categories.finite_crystals import FiniteCrystals
from sage.combinat.root_system.cartan_type import CartanType
from sage.rings.integer import Integer
from sage.rings.infinity import infinity
from sage.structure.richcmp import richcmp


class Subcrystal(UniqueRepresentation, Parent):
    """
    A subcrystal `X` of an ambient crystal `Y` is a crystal formed by taking a
    subset of `Y` and whose crystal structure is induced by `Y`.

    INPUT:

    - ``ambient`` -- the ambient crystal
    - ``contained`` -- (optional) a set (or function) which specifies when an
      element is contained in the subcrystal; the default is everything
      possible is included
    - ``generators`` -- (optional) the generators for the subcrystal; the
      default is the generators for the ambient crystal
    - ``virtualization``, ``scaling_factors`` -- (optional)
      dictionaries whose key `i` corresponds to the sets `\sigma_i`
      and `\gamma_i` respectively used to define virtual crystals; see
      :class:`~sage.combinat.crystals.virtual_crystal.VirtualCrystal`
    - ``cartan_type`` -- (optional) the Cartan type for the subcrystal; the
      default is the Cartan type for the ambient crystal
    - ``index_set`` -- (optional) the index set for the subcrystal; the
      default is the index set for the Cartan type
    - ``category`` -- (optional) the category for the subcrystal; the
      default is the :class:`~sage.categories.crystals.Crystals` category

    .. SEEALSO::

        :meth:`~sage.categories.crystals.Crystals.ParentMethods.subcrystal`

    EXAMPLES:

    We build out a subcrystal starting from an element and only going
    to the lowest weight::

        sage: B = crystals.Tableaux(['A',3], shape=[2,1])
        sage: S = B.subcrystal(generators=[B(3,1,2)], direction='lower')
        sage: S.cardinality()
        11

    Here we build out in both directions starting from an element, but we
    also have restricted ourselves to type `A_2`::

        sage: T = B.subcrystal(index_set=[1,2], generators=[B(3,1,1)])
        sage: T.cardinality()
        8
        sage: list(T)
        [[[1, 1], [3]],
         [[1, 1], [2]],
         [[1, 2], [3]],
         [[2, 2], [3]],
         [[1, 2], [2]],
         [[2, 3], [3]],
         [[1, 3], [2]],
         [[1, 3], [3]]]

    Now we take the crystal corresponding to the intersection of
    the previous two subcrystals::

        sage: U = B.subcrystal(contained=lambda x: x in S and x in T, generators=B)
        sage: list(U)
        [[[2, 3], [3]], [[1, 2], [3]], [[2, 2], [3]]]

    .. TODO::

        Include support for subcrystals which only contains certain arrows.
    """
    @staticmethod
    def __classcall_private__(cls, ambient, contained=None, generators=None,
                              virtualization=None, scaling_factors=None,
                              cartan_type=None, index_set=None, category=None):
        """
        Normalize arguments to ensure a (relatively) unique representation.

        EXAMPLES::

            sage: B = crystals.Tableaux(['A',4], shape=[2,1])
            sage: S1 = B.subcrystal(generators=(B(2,1,1), B(5,2,4)), index_set=[1,2])
            sage: S2 = B.subcrystal(generators=[B(2,1,1), B(5,2,4)], cartan_type=['A',4], index_set=(1,2))
            sage: S1 is S2
            True
        """
        if isinstance(contained, (list, tuple, set, frozenset)):
            contained = frozenset(contained)
        #elif contained in Sets():

        if cartan_type is None:
            cartan_type = ambient.cartan_type()
        else:
            cartan_type = CartanType(cartan_type)
        if index_set is None:
            index_set = cartan_type.index_set
        if generators is None:
            generators = ambient.module_generators

        category = Crystals().or_subcategory(category)
        if ambient in FiniteCrystals() or isinstance(contained, frozenset):
            category = category.Finite()

        if virtualization is not None:
            if scaling_factors is None:
                scaling_factors = {i:1 for i in index_set}
            from sage.combinat.crystals.virtual_crystal import VirtualCrystal
            return VirtualCrystal(ambient, virtualization, scaling_factors, contained,
                                  generators, cartan_type, index_set, category)
        if scaling_factors is not None:
            # virtualization must be None
            virtualization = {i:(i,) for i in index_set}
            from sage.combinat.crystals.virtual_crystal import VirtualCrystal
            return VirtualCrystal(ambient, virtualization, scaling_factors, contained,
                                  generators, cartan_type, index_set, category)

        # We need to give these as optional arguments so it unpickles correctly
        return super(Subcrystal, cls).__classcall__(cls, ambient, contained,
                                                    tuple(generato