Coverage for local/lib/python2.7/site-packages/sage/combinat/crystals/bkk_crystals.py : 98%

""" 

Benkart-Kang-Kashiwara crystals for the general-linear Lie superalgebra 

""" 

#***************************************************************************** 

# Copyright (C) 2017 Franco Saliola <saliola@gmail.com> 

# 2017 Travis Scrimshaw <tcscrims at gmail.com> 

# 2017 Anne Schilling <anne@math.ucdavis.edu> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.misc.cachefunc import cached_method 

from sage.misc.lazy_attribute import lazy_attribute 

from sage.structure.parent import Parent 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.categories.regular_supercrystals import RegularSuperCrystals 

from sage.combinat.partition import _Partitions 

from sage.combinat.root_system.cartan_type import CartanType 

from sage.combinat.crystals.letters import CrystalOfBKKLetters 

from sage.combinat.crystals.tensor_product import CrystalOfWords 

from sage.combinat.crystals.tensor_product_element import CrystalOfBKKTableauxElement 

class CrystalOfBKKTableaux(CrystalOfWords): 

r""" 

Crystal of tableaux for type `A(m|n)`. 

This is an implementation of the tableaux model of the 

Benkart-Kang-Kashiwara crystal [BKK2000]_ for the Lie 

superalgebra `\mathfrak{gl}(m+1,n+1)`. 

INPUT: 

- ``ct`` -- a super Lie Cartan type of type `A(m|n)` 

- ``shape`` -- shape specifying the highest weight; this should be 

a partition contained in a hook of height `n+1` and width `m+1` 

EXAMPLES:: 

sage: T = crystals.Tableaux(['A', [1,1]], shape = [2,1]) 

sage: T.cardinality() 

20 

""" 

@staticmethod 

def __classcall_private__(cls, ct, shape): 

""" 

Normalize input to ensure a unique representation. 

TESTS:: 

sage: crystals.Tableaux(['A', [1, 2]], shape=[2,1]) 

Crystal of BKK tableaux of shape [2, 1] of gl(2|3) 

sage: crystals.Tableaux(['A', [1, 1]], shape=[3,3,3]) 

Traceback (most recent call last): 

... 

ValueError: invalid hook shape 

""" 

ct = CartanType(ct) 

shape = _Partitions(shape) 

if len(shape) > ct.m + 1 and shape[ct.m+1] > ct.n + 1: 

raise ValueError("invalid hook shape") 

return super(CrystalOfBKKTableaux, cls).__classcall__(cls, ct, shape) 

def __init__(self, ct, shape): 

r""" 

Initialize ``self``. 

TESTS:: 

sage: T = crystals.Tableaux(['A', [1,1]], shape = [2,1]); T 

Crystal of BKK tableaux of shape [2, 1] of gl(2|2) 

sage: TestSuite(T).run() 

""" 

self._shape = shape 

self._cartan_type = ct 

m = ct.m + 1 

n = ct.n + 1 

C = CrystalOfBKKLetters(ct) 

tr = shape.conjugate() 

mg = [] 

for i,col_len in enumerate(tr): 

for j in range(col_len - m): 

mg.append(C(i+1)) 

for j in range(max(0, m - col_len), m): 

mg.append(C(-j-1)) 

mg = list(reversed(mg)) 

Parent.__init__(self, category=RegularSuperCrystals()) 

self.module_generators = (self.element_class(self, mg),) 

def _repr_(self): 

""" 

Return a string representation of ``self``. 

TESTS:: 

sage: crystals.Tableaux(['A', [1, 2]], shape=[2,1]) 

Crystal of BKK tableaux of shape [2, 1] of gl(2|3) 

""" 

m = self._cartan_type.m + 1 

n = self._cartan_type.n + 1 

return "Crystal of BKK tableaux of shape {} of gl({}|{})".format(self.shape(), m, n) 

def shape(self): 

r""" 

Return the shape of ``self``. 

EXAMPLES:: 

sage: T = crystals.Tableaux(['A', [1, 2]], shape=[2,1]) 

sage: T.shape() 

[2, 1] 

""" 

return self._shape 

def genuine_highest_weight_vectors(self, index_set=None): 

""" 

Return a tuple of genuine highest weight elements. 

A *fake highest weight vector* is one which is annihilated by 

`e_i` for all `i` in the index set, but whose weight is not 

bigger in dominance order than all other elements in the 

crystal. A *genuine highest weight vector* is a highest 

weight element that is not fake. 

EXAMPLES:: 

sage: B = crystals.Tableaux(['A', [1,1]], shape=[3,2,1]) 

sage: B.genuine_highest_weight_vectors() 

([[-2, -2, -2], [-1, -1], [1]],) 

sage: B.highest_weight_vectors() 

([[-2, -2, -2], [-1, 2], [1]], 

[[-2, -2, -2], [-1, -1], [1]], 

[[-2, -2, 2], [-1, -1], [1]]) 

""" 

if index_set is None or index_set == self.index_set(): 

return self.module_generators 

return super(CrystalOfBKKTableaux, self).genuine_highest_weight_vectors(index_set) 

class Element(CrystalOfBKKTableauxElement): 

pass