def get_initial_chain(self, highest_weight):
"""
Called internally by __init__() to construct the chain of roots
associated to the highest weight element.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.get_initial_chain(RootSystem(['A',3]).weight_space().basis()[1])
[[alpha[1], alpha[1]], [alpha[1] + alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
"""
pos_roots = self.positive_roots
tis = self.R.index_set()
tis.reverse()
cv_to_pos_root = {}
to_sort = []
for r in pos_roots:
j = highest_weight.scalar( r.associated_coroot() )
if (int(math.floor(j)) - j == Integer(0)):
j = j - Integer(1)
j = int(math.floor(j))
for k in (ellipsis_range(Integer(0),Ellipsis,j)):
cv = []
cv.append((Integer(1)/highest_weight.scalar(r.associated_coroot()))*k)
for i in tis:
cv.append((Integer(1)/highest_weight.scalar(r.associated_coroot()))*r.associated_coroot().coefficient(i))
cv_to_pos_root[ str(cv) ] = r
to_sort.append( cv )
to_sort.sort() # Note: Python sorts nested lists lexicographically by default.
lambda_chain = []
for t in to_sort:
lambda_chain.append( [ cv_to_pos_root[str(t)], cv_to_pos_root[str(t)] ] )
return lambda_chain
def fold(self, i, chain, endweight):
r"""
Returns a new chain that results from folding at position i.
Called internally by __init__().
TESTS::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C([0]).get_chain_from_subset()
([[alpha[2], -alpha[2]], [alpha[1], alpha[1]], [alpha[3], alpha[3]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]], 3/2*alpha[1] + alpha[2] + 3/2*alpha[3])
"""
s = self.parent().R.root_space().reflection(chain[i][Integer(0)])
chain[i][Integer(1)] = s(chain[i][Integer(1)])
for j in (ellipsis_range((i+Integer(1)),Ellipsis,len(chain)-Integer(1))):
pair = chain[j]
pair[Integer(0)] = s(pair[Integer(0)])
pair[Integer(1)] = s(pair[Integer(1)])
return (chain, s(endweight))
def fold(self, i, chain, endweight):
r"""
Returns a new chain that results from folding at position i.
Called internally by __init__().
TESTS::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C([0]).get_chain_from_subset()
([[alpha[2], -alpha[2]], [alpha[1], alpha[1]], [alpha[3], alpha[3]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]], 3/2*alpha[1] + alpha[2] + 3/2*alpha[3])
"""
s = self.parent().R.root_space().reflection(chain[i][Integer(0)])
chain[i][Integer(1)] = s(chain[i][Integer(1)])
for j in (ellipsis_range((i + Integer(1)), Ellipsis,
len(chain) - Integer(1))):
pair = chain[j]
pair[Integer(0)] = s(pair[Integer(0)])
pair[Integer(1)] = s(pair[Integer(1)])
return (chain, s(endweight))
def get_initial_chain(self, highest_weight):
"""
Called internally by __init__() to construct the chain of roots
associated to the highest weight element.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['A',3],[0,1,0])
sage: C.get_initial_chain(RootSystem(['A',3]).weight_space().basis()[1])
[[alpha[1], alpha[1]], [alpha[1] + alpha[2], alpha[1] + alpha[2]], [alpha[1] + alpha[2] + alpha[3], alpha[1] + alpha[2] + alpha[3]]]
"""
pos_roots = self.positive_roots
tis = self.R.index_set()
tis.reverse()
cv_to_pos_root = {}
to_sort = []
for r in pos_roots:
j = highest_weight.scalar(r.associated_coroot())
if (int(math.floor(j)) - j == Integer(0)):
j = j - Integer(1)
j = int(math.floor(j))
for k in (ellipsis_range(Integer(0), Ellipsis, j)):
cv = []
cv.append(
(Integer(1) / highest_weight.scalar(r.associated_coroot()))
* k)
for i in tis:
cv.append((Integer(1) /
highest_weight.scalar(r.associated_coroot())) *
r.associated_coroot().coefficient(i))
cv_to_pos_root[str(cv)] = r
to_sort.append(cv)
to_sort.sort(
) # Note: Python sorts nested lists lexicographically by default.
lambda_chain = []
for t in to_sort:
lambda_chain.append(
[cv_to_pos_root[str(t)], cv_to_pos_root[str(t)]])
return lambda_chain
def f(self, i, verbose=False):
r"""
Returns the action of `f_i` on self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['E',6],[1,0,0,0,0,0]) # long time (21s on sage.math, 2011)
sage: C.module_generators # long time
[[]]
sage: C([]).f(1) # long time
[0]
sage: C([]).f(1).f(3) # long time
[0, 1]
sage: C([]).f(1).f(3).f(4) # long time
[0, 1, 2]
sage: C([]).f(1).f(3).f(4).f(5) # long time
[0, 1, 2, 4]
sage: C([]).f(1).f(3).f(4).f(2) # long time
[0, 1, 2, 3]
"""
assert i in self.index_set()
# Construct the chain associated to the subset self.value
if self.chain is None:
if (str(self.value) in self.parent().chain_cache.keys()):
self.chain = self.parent().chain_cache[ str(self.value) ]
self.endweight = self.parent().endweight_cache[ str(self.value) ]
else:
(self.chain, self.endweight) = self.get_chain_from_subset()
self.parent().chain_cache[str(self.value)] = copy.deepcopy(self.chain)
self.parent().endweight_cache[str(self.value)] = copy.deepcopy(self.endweight)
ai = self.parent().R.root_space().simple_root(i)
if verbose == True:
print "( self.value, i = ", self.value, i, " ) "
print "( chain = ", self.chain, " ) "
IJp = filter( lambda j: self.chain[j][Integer(0)] == ai or self.chain[j][Integer(0)] == (-Integer(1))*ai, (ellipsis_range(Integer(0) ,Ellipsis, len(self.chain)-Integer(1))) )
if verbose == True:
print "( IJp = ", IJp, ") "
SJp = []
for j in IJp:
pair = self.chain[j]
if ( pair[Integer(0)].coefficients()[Integer(0)] < Integer(0) ):
SJp.append( -Integer(1) )
else:
SJp.append( Integer(1) )
if ( pair[Integer(1)].coefficients()[Integer(0)] < Integer(0) ):
SJp.append( -Integer(1) )
else:
SJp.append( Integer(1) )
# Incidentally, the vector at infinity in LJp is <mu(J), a_p^check>.
# This gives an alternative way to calculate the last SJp/LJp entries.
if (self.endweight.scalar( ai.associated_coroot() ) < Integer(0)):
SJp.append( -Integer(1) )
else:
SJp.append( Integer(1) )
if verbose == True:
print "( SJp = ", SJp, ") "
LJp = []
t = Integer(0)-(Integer(1)/Integer(2))
for j in (ellipsis_range(Integer(0),Ellipsis,len(SJp)-Integer(1))):
if SJp[j] == Integer(1):
t = t+(Integer(1)/Integer(2))
elif SJp[j] == -Integer(1):
t = t-(Integer(1)/Integer(2))
if ((j % Integer(2)) == Integer(0)):
LJp.append(t)
if verbose == True:
print "( LJp = ", LJp, " ) "
MJp = max(LJp)
if verbose == True:
print "( MJp = ", MJp, " ) "
if (MJp <= Integer(0)):
return None
m = LJp.index(MJp)
k = m - Integer(1)
rsubseq = []
rsubseq.extend(self.value)
if (m < len(IJp)):
rsubseq.remove(IJp[m])
rsubseq.append(IJp[k])
rsubseq.sort()
return self.parent()(rsubseq)
def e(self, i, verbose=False):
r"""
Returns the action of `e_i` on self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['E',6],[1,0,0,0,0,0]) # long time (20s on sage.math, 2011)
sage: C.module_generators # long time
[[]]
sage: C([]).f(1) # long time
[0]
sage: C([]).f(1).e(1) # long time
[]
"""
assert i in self.index_set()
# Construct the chain associated to the subset self.value
if self.chain is None:
if (str(self.value) in self.parent().chain_cache.keys()):
self.chain = self.parent().chain_cache[ str(self.value) ]
self.endweight = self.parent().endweight_cache[ str(self.value) ]
else:
(self.chain, self.endweight) = self.get_chain_from_subset()
self.parent().chain_cache[str(self.value)] = copy.deepcopy(self.chain)
self.parent().endweight_cache[str(self.value)] = copy.deepcopy(self.endweight)
ai = self.parent().R.root_space().simple_root(i)
if verbose == True:
print "( self.value, i = ", self.value, i, " ) "
print "( chain = ", self.chain, " ) "
IJp = filter( lambda j: self.chain[j][Integer(0)] == ai or self.chain[j][Integer(0)] == (-Integer(1))*ai, (ellipsis_range(Integer(0) ,Ellipsis, len(self.chain)-Integer(1))) )
if verbose == True:
print "( IJp = ", IJp, ") "
SJp = []
for j in IJp:
pair = self.chain[j]
if ( pair[Integer(0)].coefficients()[Integer(0)] < Integer(0) ):
SJp.append( -Integer(1) )
else:
SJp.append( Integer(1) )
if ( pair[Integer(1)].coefficients()[Integer(0)] < Integer(0) ):
SJp.append( -Integer(1) )
else:
SJp.append( Integer(1) )
if (self.endweight.scalar( ai.associated_coroot() ) < Integer(0)):
SJp.append( -Integer(1) )
else:
SJp.append( Integer(1) )
if verbose == True:
print "( SJp = ", SJp, ") "
LJp = []
t = Integer(0)-(Integer(1)/Integer(2))
for j in (ellipsis_range(Integer(0),Ellipsis,len(SJp)-Integer(1))):
if SJp[j] == Integer(1):
t = t+(Integer(1)/Integer(2))
elif SJp[j] == -Integer(1):
t = t-(Integer(1)/Integer(2))
if ((j % Integer(2)) == Integer(0)):
LJp.append(t)
if verbose == True:
print "( LJp = ", LJp, " ) "
MJp = max(LJp)
if verbose == True:
print "( MJp = ", MJp, " ) "
if (MJp <= LJp[ len(LJp)-1 ]):
return None
rLJp = copy.deepcopy(LJp)
rLJp.reverse()
k = rLJp.index(MJp)
k = (len(LJp)-1) - k
m = k + 1
rsubseq = []
rsubseq.extend(self.value)
rsubseq.remove(IJp[k])
if (m < len(IJp)):
rsubseq.append(IJp[m])
rsubseq.sort()
return self.parent()(rsubseq)
def f(self, i, verbose=False):
r"""
Returns the action of `f_i` on self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['E',6],[1,0,0,0,0,0]) # long time (21s on sage.math, 2011)
sage: C.module_generators # long time
[[]]
sage: C([]).f(1) # long time
[0]
sage: C([]).f(1).f(3) # long time
[0, 1]
sage: C([]).f(1).f(3).f(4) # long time
[0, 1, 2]
sage: C([]).f(1).f(3).f(4).f(5) # long time
[0, 1, 2, 4]
sage: C([]).f(1).f(3).f(4).f(2) # long time
[0, 1, 2, 3]
"""
assert i in self.index_set()
# Construct the chain associated to the subset self.value
if self.chain is None:
if (str(self.value) in self.parent().chain_cache.keys()):
self.chain = self.parent().chain_cache[str(self.value)]
self.endweight = self.parent().endweight_cache[str(self.value)]
else:
(self.chain, self.endweight) = self.get_chain_from_subset()
self.parent().chain_cache[str(self.value)] = copy.deepcopy(
self.chain)
self.parent().endweight_cache[str(self.value)] = copy.deepcopy(
self.endweight)
ai = self.parent().R.root_space().simple_root(i)
if verbose == True:
print "( self.value, i = ", self.value, i, " ) "
print "( chain = ", self.chain, " ) "
IJp = filter(
lambda j: self.chain[j][Integer(0)] == ai or self.chain[j][Integer(
0)] == (-Integer(1)) * ai,
(ellipsis_range(Integer(0), Ellipsis,
len(self.chain) - Integer(1))))
if verbose == True:
print "( IJp = ", IJp, ") "
SJp = []
for j in IJp:
pair = self.chain[j]
if (pair[Integer(0)].coefficients()[Integer(0)] < Integer(0)):
SJp.append(-Integer(1))
else:
SJp.append(Integer(1))
if (pair[Integer(1)].coefficients()[Integer(0)] < Integer(0)):
SJp.append(-Integer(1))
else:
SJp.append(Integer(1))
# Incidentally, the vector at infinity in LJp is <mu(J), a_p^check>.
# This gives an alternative way to calculate the last SJp/LJp entries.
if (self.endweight.scalar(ai.associated_coroot()) < Integer(0)):
SJp.append(-Integer(1))
else:
SJp.append(Integer(1))
if verbose == True:
print "( SJp = ", SJp, ") "
LJp = []
t = Integer(0) - (Integer(1) / Integer(2))
for j in (ellipsis_range(Integer(0), Ellipsis, len(SJp) - Integer(1))):
if SJp[j] == Integer(1):
t = t + (Integer(1) / Integer(2))
elif SJp[j] == -Integer(1):
t = t - (Integer(1) / Integer(2))
if ((j % Integer(2)) == Integer(0)):
LJp.append(t)
if verbose == True:
print "( LJp = ", LJp, " ) "
MJp = max(LJp)
if verbose == True:
print "( MJp = ", MJp, " ) "
if (MJp <= Integer(0)):
return None
m = LJp.index(MJp)
k = m - Integer(1)
rsubseq = []
rsubseq.extend(self.value)
if (m < len(IJp)):
rsubseq.remove(IJp[m])
rsubseq.append(IJp[k])
rsubseq.sort()
return self.parent()(rsubseq)
def e(self, i, verbose=False):
r"""
Returns the action of `e_i` on self.
EXAMPLES::
sage: C = ClassicalCrystalOfAlcovePaths(['E',6],[1,0,0,0,0,0]) # long time (20s on sage.math, 2011)
sage: C.module_generators # long time
[[]]
sage: C([]).f(1) # long time
[0]
sage: C([]).f(1).e(1) # long time
[]
"""
assert i in self.index_set()
# Construct the chain associated to the subset self.value
if self.chain is None:
if (str(self.value) in self.parent().chain_cache.keys()):
self.chain = self.parent().chain_cache[str(self.value)]
self.endweight = self.parent().endweight_cache[str(self.value)]
else:
(self.chain, self.endweight) = self.get_chain_from_subset()
self.parent().chain_cache[str(self.value)] = copy.deepcopy(
self.chain)
self.parent().endweight_cache[str(self.value)] = copy.deepcopy(
self.endweight)
ai = self.parent().R.root_space().simple_root(i)
if verbose == True:
print "( self.value, i = ", self.value, i, " ) "
print "( chain = ", self.chain, " ) "
IJp = filter(
lambda j: self.chain[j][Integer(0)] == ai or self.chain[j][Integer(
0)] == (-Integer(1)) * ai,
(ellipsis_range(Integer(0), Ellipsis,
len(self.chain) - Integer(1))))
if verbose == True:
print "( IJp = ", IJp, ") "
SJp = []
for j in IJp:
pair = self.chain[j]
if (pair[Integer(0)].coefficients()[Integer(0)] < Integer(0)):
SJp.append(-Integer(1))
else:
SJp.append(Integer(1))
if (pair[Integer(1)].coefficients()[Integer(0)] < Integer(0)):
SJp.append(-Integer(1))
else:
SJp.append(Integer(1))
if (self.endweight.scalar(ai.associated_coroot()) < Integer(0)):
SJp.append(-Integer(1))
else:
SJp.append(Integer(1))
if verbose == True:
print "( SJp = ", SJp, ") "
LJp = []
t = Integer(0) - (Integer(1) / Integer(2))
for j in (ellipsis_range(Integer(0), Ellipsis, len(SJp) - Integer(1))):
if SJp[j] == Integer(1):
t = t + (Integer(1) / Integer(2))
elif SJp[j] == -Integer(1):
t = t - (Integer(1) / Integer(2))
if ((j % Integer(2)) == Integer(0)):
LJp.append(t)
if verbose == True:
print "( LJp = ", LJp, " ) "
MJp = max(LJp)
if verbose == True:
print "( MJp = ", MJp, " ) "
if (MJp <= LJp[len(LJp) - 1]):
return None
rLJp = copy.deepcopy(LJp)
rLJp.reverse()
k = rLJp.index(MJp)
k = (len(LJp) - 1) - k
m = k + 1
rsubseq = []
rsubseq.extend(self.value)
rsubseq.remove(IJp[k])
if (m < len(IJp)):
rsubseq.append(IJp[m])
rsubseq.sort()
return self.parent()(rsubseq)
