Sharedfinal.ipynbOpen in CoCalc
import __future__
from Weight import Weight
from PartitionTree import PartitionTree
def Eval(string):
return eval(compile(str(string), '<string>', 'eval', __future__.division.compiler_flag))

def getBasisChange(name):
simples = RootSystem(name).ambient_space().simple_roots()
bChange = ([Eval((str(x).replace("(", "[").replace(")", "]"))) for x in simples])
for i in range(len(bChange), len(bChange[0])):
bChange.append([1 if j==i else 0 for j in range(0, len(bChange[0]))])
return matrix(bChange).transpose().inverse()

def geometricSumForPartition(positive_root, translations, q_analog):
x = 1
for i in range(0, len(positive_root)):
for j in range(0, positive_root[i]):
x = x * translations["A" + str(i+1)]
return 1/(1 - x) if not q_analog else 1/(1 -q*x)

def calculatePartition(name, weight, positive_roots = [], translations = {}, q_analog = False):
if positive_roots == []:
bChange = getBasisChange(name)
positive_roots = [vector(list(Eval(x))) for x in RootSystem(name).ambient_space().positive_roots()]
positive_roots = [bChange * x for x in positive_roots]

if translations == {}:
s = ''
if q_analog:
s = 'q, '
s += 'A1'
for i in range(1, len(weight)):
s += ', A' + str(i + 1)
variables = var(s)
for i in range(0, len(weight)):
translations["A" + str(i+1)] = eval("A" + str(i+1))

termsForSum = [geometricSumForPartition(list(x), translations, q_analog) for x in positive_roots]
for x in termsForSum:

for i in range(0, len(weight)):

def findAltSet(name, lamb = None, mu = None):
# initialize constants and vector space for the lie algebra
lie_algebra = RootSystem(name).ambient_space()
weyl_group = WeylGroup(name, prefix = "s")
simples = weyl_group.gens()

altset = [weyl_group.one()]

# used to change the basis from the standard basis of R^n to simple roots
changeBasis = getBasisChange(name)

# if lambda is not specified, the highest root is used
if lamb == None:
lamb = lie_algebra.highest_root()

# if mu is not specified, 0 vector is used
if mu == None:
mu = weyl_group.domain()([0 for i in range(0, len(lie_algebra.simple_roots()))])

# check to see if the alt set is the empty set
init = (lamb + mu)
init = changeBasis * vector(list(Eval(init)))
init = Weight(init)

if init.isNegative():
return []

rho = lie_algebra.rho()
length = len(altset)
i=0
while i < length:
for simple in simples:
if ((altset[i] == simple)or (altset[i] == altset[i] * simple)):
continue
res = (altset[i]*simple).action(lamb + rho) - (rho + mu)
res = changeBasis * vector(list(Eval(res)))
res = Weight(res)

if not (res.isNegative() or res.hasFraction()):
if not (altset[i]*simple in altset):
altset.append(altset[i]*simple)
length += 1
i+=1

return altset

def calculateMultiplicity(name, lamb = None, mu = None, q_analog = False):
mult = 0
lie_algebra = RootSystem(name).ambient_space()
weyl_group = WeylGroup(name, prefix = "s")

# used to change the basis from the standard basis of R^n to simple roots
changeBasis = getBasisChange(name)

positive_roots = [vector(list(Eval(x))) for x in RootSystem(name).ambient_space().positive_roots()]
positive_roots = [getBasisChange(name) * x for x in positive_roots]

# if lambda is not specified, the highest root is used
if lamb == None:
lamb = lie_algebra.highest_root()
else:
while not len(lamb) == changeBasis.ncols():
lamb.append(0)
lamb = changeBasis.inverse() * vector(lamb)
lamb = weyl_group.domain()(list(eval(str(lamb))))

# if mu is not specified, 0 vector is used
if mu == None:
mu = weyl_group.domain()([0 for i in range(0, len(lie_algebra.simple_roots()))])
else:
while not len(mu) == changeBasis.ncols():
mu.append(0)
mu = changeBasis.inverse() * vector(mu)
mu = weyl_group.domain()(list(eval(str(mu))))

rho = lie_algebra.rho()
altset = findAltSet(name, lamb, mu)
#print(changeBasis * vector(list(Eval(lamb))))
for elm in altset:
# expression in partition function
res = elm.action(lamb + rho) - (mu + rho)

#change basis from standard basis to simple roots
res = vector(list(Eval(res)))
res = changeBasis * res

term = calculatePartition(name, list(res), positive_roots, q_analog=q_analog)

term *= (-1)**elm.length()
mult += term

return mult

def printPartitions(name, weight, tex):
lie_algebra = RootSystem(name).ambient_space()

# used to change the basis from the standard basis of R^n to simple roots
changeBasis = getBasisChange(name)

positive_roots = [vector(list(Eval(x))) for x in RootSystem(name).ambient_space().positive_roots()]
positive_roots = [getBasisChange(name) * x for x in positive_roots]
latex_roots = ["".join([str(root[j] if root[j] != 1 else "") + "\\alpha_" + str(j+1)+"+" if root[j] != 0 else "" for j in range(0, len(root))]) for root in positive_roots]
latex_roots = [root[0:len(root)-1] for root in latex_roots]
weight_positive_roots = [Weight(list(root)) for root in positive_roots]
weight = Weight(weight)
tree = PartitionTree(weight, weight_positive_roots, 0, 0)

partitions = []
tree.getPartitions(partitions)

output = open(tex, "w")
output.write("\\documentclass{article}\n\\begin{document}\n")
for partition in partitions:
string = ""
for i in range(len(partition)):
coeff = partition[i]
if coeff > 1:
string += str(coeff) + "(" + latex_roots[i] + ") +"
elif coeff > 0:
string += "(" + latex_roots[i] + ")+"
string = "$"+ string[0:len(string)-1] string += "$\\\\\\\\\n\n"
output.write(string)
output.write("\\end{document}")
output.close()
print "Done"

#first argument is the name of the lie algebra
#second argument is lambda, an array that has the coefficients of the simple roots in order of subscript
#third argument is mu, given in the same way as lambda
#fourth argument is a boolean flag, so True/False, that determines whether to do the q-analog or not
#you can ommit any of the last three arguments and a default will be used
#default for lambda is the highest positive root
#default for mu is the 0 weight
#default for the q analog is false
#if you want to use a subset of the last three, you can set them by name.
#set lambda with lamb = ..., mu with mu = ..., and q-analog with q_analog = True/False
#Below are examples
#calculateMultiplicity("A3", [3,3,3], [2,2,2], True)
#calculateMultiplicity("A3", [3,3,3], q_analog=True)
#calculateMultiplicity("B3",lamb=[1,2,2], mu=[0,0], q_analog=False)
#printPartitions("A1", [5], "output.tex")
#findAltSet("A2")
calculatePartition("A6", [4,4,4,4,4,4],q_analog=False)

25725
calculatePartition("A2", [5,5],q_analog=False)

--------------------------------------------------------------------------- KeyError Traceback (most recent call last) <ipython-input-3-6856cb4a01ec> in <module>() ----> 1 calculatePartition("A2", [Integer(5),Integer(5)],q_analog=False) <ipython-input-2-dafcda968526> in calculatePartition(name, weight, positive_roots, translations, q_analog) 36 translations["A" + str(i+Integer(1))] = eval("A" + str(i+Integer(1))) 37 ---> 38 termsForSum = [geometricSumForPartition(list(x), translations, q_analog) for x in positive_roots] 39 answer = Integer(1) 40 for x in termsForSum: <ipython-input-2-dafcda968526> in geometricSumForPartition(positive_root, translations, q_analog) 16 for i in range(Integer(0), len(positive_root)): 17 for j in range(Integer(0), positive_root[i]): ---> 18 x = x * translations["A" + str(i+Integer(1))] 19 return Integer(1)/(Integer(1) - x) if not q_analog else Integer(1)/(Integer(1) -q*x) 20 KeyError: 'A2' 














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