p.harris.ipynb

Kernel:

import __future__
from Weight import Weight
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]
    answer = 1
    for x in termsForSum:
        answer *= x
        
    for i in range(0, len(weight)):
        answer = answer.series(translations["A" + str(i+1)], weight[i] + 1).truncate()
        answer = answer.coefficient(translations["A" + str(i+1)], weight[i])

    answer = answer.expand()
    return answer
    
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()
    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()
        
    # check to see if the alt set is the empty set
    init = (lamb + rho) - (rho + mu)
    init = changeBasis * vector(list(Eval(init)))
    init = Weight(init)
    
    if init.isNegative() or init.hasFraction():
        return []
        
    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

#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("A3", mu=[2,2,2], q_analog=True)
x = calculatePartition("A4", [4,4,4,4], q_analog=False)
print(x)
#findAltSet("D4")

Error: No spec available for sage-7.4