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r"""
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Weyl Groups
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"""
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- #* ****************************************************************************
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+ # ****************************************************************************
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# Copyright (C) 2009 Nicolas M. Thiery <nthiery at users.sf.net>
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#
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# Distributed under the terms of the GNU General Public License (GPL)
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- # http ://www.gnu.org/licenses/
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- #* *****************************************************************************
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+ # https ://www.gnu.org/licenses/
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+ # *****************************************************************************
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from sage .misc .cachefunc import cached_method , cached_in_parent_method
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from sage .misc .lazy_import import LazyImport
@@ -315,7 +315,7 @@ def length(x):
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from sage .graphs .digraph import DiGraph
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return DiGraph (visited ,
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- name = "Parabolic Quantum Bruhat Graph of %s for nodes %s" % (self , index_set ),
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+ name = "Parabolic Quantum Bruhat Graph of %s for nodes %s" % (self , index_set ),
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format = "dict_of_dicts" ,
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data_structure = "static_sparse" )
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@@ -493,13 +493,13 @@ def stanley_symmetric_function_as_polynomial(self, max_length=None):
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W = self .parent ()
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pieri_factors = W .pieri_factors ()
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from sage .rings .rational_field import QQ
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- R = QQ [',' .join ('x%s' % l for l in range (1 ,pieri_factors .max_length ()+ 1 ))]
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+ R = QQ [',' .join ('x%s' % l for l in range (1 , pieri_factors .max_length ()+ 1 ))]
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x = R .gens ()
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if self .is_one ():
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return R .one ()
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return R (sum (2 ** (pieri_factors .stanley_symm_poly_weight (u ))* x [u .length ()- 1 ] * v .stanley_symmetric_function_as_polynomial (max_length = u .length ())
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- for (u ,v ) in self .left_pieri_factorizations (max_length )
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+ for (u , v ) in self .left_pieri_factorizations (max_length )
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if u != W .one ()))
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def stanley_symmetric_function (self ):
@@ -747,7 +747,7 @@ def bruhat_lower_covers_coroots(self):
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[(s1*s2*s1, alphacheck[1] + alphacheck[2] + alphacheck[3]),
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(s3*s2*s1, alphacheck[2]), (s3*s1*s2, alphacheck[1])]
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"""
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- return [(x [0 ],x [1 ].reflection_to_coroot ())
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+ return [(x [0 ], x [1 ].reflection_to_coroot ())
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for x in self .bruhat_lower_covers_reflections ()]
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def bruhat_upper_covers_coroots (self ):
@@ -770,7 +770,7 @@ def bruhat_upper_covers_coroots(self):
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(s3*s4*s1*s2*s1, alphacheck[4]),
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(s4*s3*s1*s2*s1, alphacheck[1] + alphacheck[2] + alphacheck[3] + alphacheck[4])]
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"""
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- return [(x [0 ],x [1 ].reflection_to_coroot ())
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+ return [(x [0 ], x [1 ].reflection_to_coroot ())
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for x in self .bruhat_upper_covers_reflections ()]
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def quantum_bruhat_successors (self , index_set = None , roots = False , quantum_only = False ):
@@ -840,12 +840,12 @@ def quantum_bruhat_successors(self, index_set=None, roots=False, quantum_only=Fa
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wrc = wr .coset_representative (index_set )
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if wrc == wr and wr .length () == w_length_plus_one and not quantum_only :
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if roots :
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- successors .append ((wr ,alpha ))
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+ successors .append ((wr , alpha ))
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else :
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successors .append (wr )
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elif alpha .quantum_root () and wrc .length () == w_length_plus_one - lattice .nonparabolic_positive_root_sum (index_set ).scalar (alpha .associated_coroot ()):
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if roots :
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- successors .append ((wrc ,alpha ))
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+ successors .append ((wrc , alpha ))
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else :
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successors .append (wrc )
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return successors
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