def to_matrix(self):
"""
Return a matrix of ``self``.
The colors are mapped to roots of unity.
EXAMPLES::
sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t*s2; x.one_line_form()
[(1, 2), (0, 1), (0, 3)]
sage: M = x.to_matrix(); M
[ 0 1 0]
[zeta4 0 0]
[ 0 0 1]
The matrix multiplication is in the *opposite* order::
sage: M == s2.to_matrix()*t.to_matrix()*s2.to_matrix()*s1.to_matrix()
True
"""
Cp = CyclotomicField(self.parent()._m)
g = Cp.gen()
D = diagonal_matrix(Cp, [g**i for i in self._colors])
return self._perm.to_matrix() * D
def _calcMatrixTrans(calc, tS, tT, lS, lT): """ See `calcMatrixTrans()` for the main documentation. This is the lower-level function which is wrapped through `persistent_cache`. This makes the cache more flexible about different parameters `R` to `calcMatrixTrans()`. """ try: ms = calc.calcMatrixTrans(tS, tT, lS, lT) except Exception: print (calc.params, calc.curlS, tS, tT, lS, lT) raise # Each matrix is for a zeta**i factor, where zeta is the n-th root of unity. # And n = calc.matrixCountTrans. assert len(ms) == calc.matrixCountTrans order = len(ms) K = CyclotomicField(order) zeta = K.gen() Kcoords = zeta.coordinates_in_terms_of_powers() assert len(K.power_basis()) == K.degree() new_ms = [matrix(QQ, ms[0].nrows(), ms[0].ncols()) for i in range(K.degree())] for l in range(order): coords = Kcoords(zeta**l) for i,m in enumerate(coords): new_ms[i] += ms[l] * m ms = new_ms denom = calc.matrixRowDenomTrans denom, ms = reduceNRow(denom=denom, mats=ms) return denom, order, ms
def to_matrix(self):
"""
Return a matrix of ``self``.
The colors are mapped to roots of unity.
EXAMPLES::
sage: C = ColoredPermutations(4, 3)
sage: s1,s2,t = C.gens()
sage: x = s1*s2*t*s2; x.one_line_form()
[(1, 2), (0, 1), (0, 3)]
sage: M = x.to_matrix(); M
[ 0 1 0]
[zeta4 0 0]
[ 0 0 1]
The matrix multiplication is in the *opposite* order::
sage: M == s2.to_matrix()*t.to_matrix()*s2.to_matrix()*s1.to_matrix()
True
"""
Cp = CyclotomicField(self.parent()._m)
g = Cp.gen()
D = diagonal_matrix(Cp, [g ** i for i in self._colors])
return self._perm.to_matrix() * D
def eis_phipsi(phi, psi, k, prec=10, t=1, cmplx=False):
r"""
Return Fourier expansion of Eisenstein series at the cusp oo.
INPUT:
- ``phi`` -- Dirichlet character.
- ``psi`` -- Dirichlet character.
- ``k`` -- integer, the weight of the Eistenstein series.
- ``prec`` -- integer (default: 10).
- ``t`` -- integer (default: 1).
OUTPUT:
The Fourier expansion of the Eisenstein series $E_k^{\phi,\psi, t}$ (as
defined by [Diamond-Shurman]).
EXAMPLES:
sage: phi = DirichletGroup(3)[1]
sage: psi = DirichletGroup(5)[1]
sage: E = eisenstein_series_at_inf(phi, psi, 4)
"""
N1, N2 = phi.level(), psi.level()
N = N1 * N2
#The Fourier expansion of the Eisenstein series at infinity is in the field Q(zeta_Ncyc)
Ncyc = lcm([euler_phi(N1), euler_phi(N2)])
if cmplx == True:
CC = ComplexField(53)
pi = ComplexField().pi()
I = ComplexField().gen()
R = CC
zeta = CC(exp(2 * pi * I / Ncyc))
else:
R = CyclotomicField(Ncyc)
zeta = R.zeta(Ncyc)
phi, psi = phi.base_extend(R), psi.base_extend(R)
Q = PowerSeriesRing(R, 'q')
q = Q.gen()
s = O(q**prec)
#Weight 2 with trivial characters is calculated separately
if k == 2 and phi.conductor() == 1 and psi.conductor() == 1:
if t == 1:
raise TypeError('E_2 is not a modular form.')
s = 1 / 24 * (t - 1)
for m in srange(1, prec):
for n in srange(1, prec / m + 1):
s += n * (q**(m * n) - t * q**(m * n * t))
return s + O(q**prec)
if psi.level() == 1 and k == 1:
s -= phi.bernoulli(k) / k
elif phi.level() == 1:
s -= psi.bernoulli(k) / k
for m in srange(1, prec / t):
for n in srange(1, prec / t / m + 1):
s += 2 * phi(m) * psi(n) * n**(k - 1) * q**(m * n * t)
return s + O(q**prec)
def to_cyclotomic_field(self, R=None):
r"""
Return this element as an element of a cyclotomic field.
EXAMPLES::
sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(3).to_cyclotomic_field()
zeta3
sage: UCF.gen(3,2).to_cyclotomic_field()
-zeta3 - 1
sage: CF = CyclotomicField(5)
sage: CF(E(5)) # indirect doctest
zeta5
sage: CF = CyclotomicField(7)
sage: CF(E(5)) # indirect doctest
Traceback (most recent call last):
...
TypeError: Cannot coerce zeta5 into Cyclotomic Field of order 7 and
degree 6
sage: CF = CyclotomicField(10)
sage: CF(E(5)) # indirect doctest
zeta10^2
Matrices are correctly dealt with::
sage: M = Matrix(UCF,2,[E(3),E(4),E(5),E(6)]); M
[ E(3) E(4)]
[ E(5) -E(3)^2]
sage: Matrix(CyclotomicField(60),M) # indirect doctest
[zeta60^10 - 1 zeta60^15]
[ zeta60^12 zeta60^10]
Using a non-standard embedding::
sage: CF = CyclotomicField(5,embedding=CC(exp(4*pi*i/5)))
sage: x = E(5)
sage: CC(x)
0.309016994374947 + 0.951056516295154*I
sage: CC(CF(x))
0.309016994374947 + 0.951056516295154*I
"""
from sage.rings.number_field.number_field import CyclotomicField
k = self._obj.Conductor().sage()
Rcan = CyclotomicField(k)
if R is None:
R = Rcan
obj = self._obj
if obj.IsRat():
return R(obj.sage())
zeta = Rcan.gen()
coeffs = obj.CoeffsCyc(k).sage()
return R(sum(coeffs[a] * zeta**a for a in range(1, k)))
def to_cyclotomic_field(self, R=None):
r"""
Return this element as an element of a cyclotomic field.
EXAMPLES::
sage: UCF = UniversalCyclotomicField()
sage: UCF.gen(3).to_cyclotomic_field()
zeta3
sage: UCF.gen(3,2).to_cyclotomic_field()
-zeta3 - 1
sage: CF = CyclotomicField(5)
sage: CF(E(5)) # indirect doctest
zeta5
sage: CF = CyclotomicField(7)
sage: CF(E(5)) # indirect doctest
Traceback (most recent call last):
...
TypeError: Cannot coerce zeta5 into Cyclotomic Field of order 7 and
degree 6
sage: CF = CyclotomicField(10)
sage: CF(E(5)) # indirect doctest
zeta10^2
Matrices are correctly dealt with::
sage: M = Matrix(UCF,2,[E(3),E(4),E(5),E(6)]); M
[ E(3) E(4)]
[ E(5) -E(3)^2]
sage: Matrix(CyclotomicField(60),M) # indirect doctest
[zeta60^10 - 1 zeta60^15]
[ zeta60^12 zeta60^10]
Using a non-standard embedding::
sage: CF = CyclotomicField(5,embedding=CC(exp(4*pi*i/5)))
sage: x = E(5)
sage: CC(x)
0.309016994374947 + 0.951056516295154*I
sage: CC(CF(x))
0.309016994374947 + 0.951056516295154*I
"""
from sage.rings.number_field.number_field import CyclotomicField
k = self._obj.Conductor().sage()
Rcan = CyclotomicField(k)
if R is None:
R = Rcan
obj = self._obj
if obj.IsRat():
return R(obj.sage())
zeta = Rcan.gen()
coeffs = obj.CoeffsCyc(k).sage()
return R(sum(coeffs[a] * zeta**a for a in range(1,k)))
def _rank(self, K) :
if K is QQ or K in NumberFields() :
return len(_jacobi_forms_by_taylor_expansion_coords(self.__index, self.__weight, 0))
## This is the formula used by Poor and Yuen in Paramodular cusp forms
if self.__weight == 2 :
delta = len(self.__index.divisors()) // 2 - 1
else :
delta = 0
return sum( ModularForms(1, self.__weight + 2 * j).dimension() + j**2 // (4 * self.__index)
for j in xrange(self.__index + 1) ) \
+ delta
## This is the formula given by Skoruppa in
## Jacobi forms of critical weight and Weil representations
##FIXME: There is some mistake here
if self.__weight % 2 != 0 :
## Otherwise the space X(i**(n - 2 k)) is different
## See: Skoruppa, Jacobi forms of critical weight and Weil representations
raise NotImplementedError
m = self.__index
K = CyclotomicField(24 * m, 'zeta')
zeta = K.gen(0)
quadform = lambda x : 6 * x**2
bilinform = lambda x,y : quadform(x + y) - quadform(x) - quadform(y)
T = diagonal_matrix([zeta**quadform(i) for i in xrange(2*m)])
S = sum(zeta**(-quadform(x)) for x in xrange(2 * m)) / (2 * m) \
* matrix([[zeta**(-bilinform(j,i)) for j in xrange(2*m)] for i in xrange(2*m)])
subspace_matrix_1 = matrix( [ [1 if j == i or j == 2*m - i else 0 for j in xrange(m + 1) ]
for i in xrange(2*m)] )
subspace_matrix_2 = zero_matrix(ZZ, m + 1, 2*m)
subspace_matrix_2.set_block(0,0,identity_matrix(m+1))
T = subspace_matrix_2 * T * subspace_matrix_1
S = subspace_matrix_2 * S * subspace_matrix_1
sqrt3 = (zeta**(4*m) - zeta**(-4*m)) * zeta**(-6*m)
rank = (self.__weight - 1/2 - 1) / 2 * (m + 1) \
+ 1/8 * ( zeta**(3*m * (2*self.__weight - 1)) * S.trace()
+ zeta**(3*m * (1 - 2*self.__weight)) * S.trace().conjugate() ) \
+ 2/(3*sqrt3) * ( zeta**(4 * m * self.__weight) * (S*T).trace()
+ zeta**(-4 * m * self.__weight) * (S*T).trace().conjugate() ) \
- sum((j**2 % (m+1))/(m+1) -1/2 for j in range(0,m+1))
if self.__weight > 5 / 2 :
return rank
else :
raise NotImplementedError
raise NotImplementedError
def __init__(self, n):
"""
Initialize ``self``.
EXAMPLES::
sage: G = groups.matrix.BinaryDihedral(4)
sage: TestSuite(G).run()
"""
self._n = n
if n % 2 == 0:
R = CyclotomicField(2 * n)
zeta = R.gen()
i = R.gen()**(n // 2)
else:
R = CyclotomicField(4 * n)
zeta = R.gen()**2
i = R.gen()**n
MS = MatrixSpace(R, 2)
zero = R.zero()
gens = [MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero])]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self,
ZZ(2),
R,
gap_group,
category=Groups().Finite())
def toLowerCyclBase(ms, old_order, new_order): """ Let's K_old = CyclotomicField(old_order) , K_new = CyclotomicField(new_order) . We transform the matrices in power base from `K_old` to `K_new`. The way this is implemented works only if `old_order` is a multiple of `new_order`. INPUT: - `ms` -- A list of matrices where every matrix is a factor to `zeta_old**i` where `zeta_old = K_old.gen()` and `len(ms) == K_old.degree()`. - `old_order` -- The order of `K_old`. - `new_order` -- The order of `K_new`. OUTPUT: - A list of matrices `new_ms` where every matrix is a factor to `zeta_new**i` where `zeta_new = K_new.gen()` and `len(new_ms) == K_new.degree()`. """ assert isinstance(ms, list) # list of matrices assert old_order % new_order == 0 K_old = CyclotomicField(old_order) old_degree = int(ZZ(K_old.degree())) K_new = CyclotomicField(new_order) new_degree = int(ZZ(K_new.degree())) assert old_degree % new_degree == 0 assert len(ms) == old_degree new_ms = [None] * new_degree for i in range(old_degree): i2,rem = divmod(i, old_degree / new_degree) if rem == 0: new_ms[i2] = ms[i] else: if ms[i] != 0: return None return new_ms
def reduce_numberfield(a,K):
r"""
INPUT:
a - an element that has the method minpoly. One should multiply a, so it becomes integral, otherwise the result might not be correct.
K - number field, such that a is in K. If d is a number we set K = Q(zeta_d)
OUTPUT:
A representation of a in terms of zeta_d
EXAMPLE:
sage: zeta294 = CyclotomicField(294).gen()
sage: a = 2/7*zeta294^77 - 6/7*zeta294^63 - 8/7*zeta294^56 + 5/7*zeta294^42 - 2/7*zeta294^28 - 4/7*zeta294^21 + 8/7*zeta294^7 + 3/7
sage: reduce_cyclotomic(a*7,7)
-2*zeta7^5 - 4*zeta7^4 - 6*zeta7^3 - 8*zeta7^2 - 3*zeta7 - 5
"""
if K in ZZ:
K = CyclotomicField(K)
R = PolynomialRing(K, 'X')
X = R.gen()
candidates = [-p[0][0] for p in R(a.minpoly()).factor()]
distances = [abs(CC(x)-CC(a)) for x in candidates]
if min(distances)>.001:
return 'Are you sure that a is in {}?'.format(K)
return candidates[distances.index(min(distances))]
def hurwitz_hat(d, N, l, zetaN=None):
r"""
Computes the function zeta_hat(d/N, l) from Brunaults paper "Regulateurs
modulaires via la methode de Rogers--Zudilin" as an element of Q(zetaN)
at a non-positive integer l. We use the formula
sum_{x in Z/NZ} zetaN**(xu) zeta(x/N, s) = N**s zeta_hat(u/N, s).
INPUT:
- d - int
- N - int, positive int
- l - int, non-positive integer
OUTPUT:
- an element of the ring Q(zetaN), equal to zeta_hat(d/N, l)
"""
if zetaN == None:
zetaN = CyclotomicField(N).gen()
d = d % N
k = 1 - l
if k < 1:
raise ValueError, 'k has to be positive'
if k == 1:
s = -QQ(1) / QQ(
2) # This is the term in the sum corresponding to x = 0
s += sum([
zetaN**(x * d) *
(QQ(1) / QQ(2) - QQ(d) / QQ(N) + floor(QQ(d) / QQ(N)))
for x in range(1, N)
])
else:
s = sum([
-zetaN**(x * d) *
(ber_pol(QQ(x) / QQ(N) - floor(QQ(x) / QQ(N)), k)) / k
for x in range(N)
])
return N**(k - 1) * s
def __init__(self, ray_class_group, base_ring):
names = tuple([ "chi%i"%i for i in range(ray_class_group.ngens()) ])
if base_ring is None:
from sage.rings.number_field.number_field import CyclotomicField
base_ring = CyclotomicField(LCM(ray_class_group.gens_orders()))
DualAbelianGroup_class.__init__(self, ray_class_group, names, base_ring)
""" ray_class_group accessible as self.group() """
def idft(self):
"""
A discrete inverse Fourier transform. Only works over `\QQ`.
EXAMPLES::
sage: J = range(5)
sage: A = [ZZ(1) for i in J]
sage: s = IndexedSequence(A,J)
sage: fs = s.dft(); fs
Indexed sequence: [5, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4]
sage: it = fs.idft(); it
Indexed sequence: [1, 1, 1, 1, 1]
indexed by [0, 1, 2, 3, 4]
sage: it == s
True
"""
F = self.base_ring() ## elements must be coercible into QQ(zeta_N)
J = self.index_object() ## must be = range(N)
N = len(J)
S = self.list()
zeta = CyclotomicField(N).gen()
iFT = [sum([S[i]*zeta**(-i*j) for i in J]) for j in J]
if not(J[0] in ZZ) or F.base_ring().fraction_field() != QQ:
raise NotImplementedError("Sorry this type of idft is not implemented yet.")
return IndexedSequence(iFT,J)*(Integer(1)/N)
def eis_H(a, b, N, k, Q=None, t=1, prec=10):
if Q == None:
Q = PowerSeriesRing(CyclotomicField(N), 'q')
R = Q.base_ring()
zetaN = R.zeta(N)
q = Q.gen()
a = ZZ(a % N)
b = ZZ(b % N)
s = 0
if k == 1:
if a == 0 and not b == 0:
s = -QQ(1) / QQ(2) * (1 + zetaN**b) / (1 - zetaN**b)
elif b == 0 and not a == 0:
s = -QQ(1) / QQ(2) * (1 + zetaN**a) / (1 - zetaN**a)
elif a != 0 and b != 0:
s = -QQ(1) / QQ(2) * ((1 + zetaN**a) / (1 - zetaN**a) +
(1 + zetaN**b) / (1 - zetaN**b))
elif k > 1:
s = hurwitz_hat(-b, N, 1 - k, zetaN)
for m in srange(1, prec / t):
for n in srange(1, prec / t / m + 1):
s += (zetaN**(-a * m - b * n) +
(-1)**k * zetaN**(a * m + b * n)) * n**(k - 1) * q**(m * n)
return s + O(q**floor(prec))
def __init__(self, n):
"""
Initialize ``self``.
EXAMPLES::
sage: G = groups.matrix.BinaryDihedral(4)
sage: TestSuite(G).run()
"""
self._n = n
if n % 2 == 0:
R = CyclotomicField(2*n)
zeta = R.gen()
i = R.gen()**(n//2)
else:
R = CyclotomicField(4*n)
zeta = R.gen()**2
i = R.gen()**n
MS = MatrixSpace(R, 2)
zero = R.zero()
gens = [ MS([zeta, zero, zero, ~zeta]), MS([zero, i, i, zero]) ]
from sage.libs.gap.libgap import libgap
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
FinitelyGeneratedMatrixGroup_gap.__init__(self, ZZ(2), R, gap_group, category=Groups().Finite())
def __init__(self, ray_class_group, base_ring):
names = tuple(["chi%i" % i for i in range(ray_class_group.ngens())])
if base_ring is None:
from sage.rings.number_field.number_field import CyclotomicField
# FIXME: following is deprecated starting from sage 7.1
# replace by from sage.arith.all import LCM
from sage.rings.arith import LCM
base_ring = CyclotomicField(LCM(ray_class_group.gens_orders()))
DualAbelianGroup_class.__init__(self, ray_class_group, names,
base_ring)
""" ray_class_group accessible as self.group() """
def __init__(self, G, names="X", bse_ring=None):
"""
If G has invariants invs = [n1,...,nk] then the default base_ring
is CyclotoomicField(N), where N = LCM(n1,...,nk).
"""
if bse_ring == None:
base_ring = CyclotomicField(LCM(G.invariants()))
else:
base_ring = bse_ring
self.__group = G
self._assign_names(names)
self._base_ring = base_ring
def extract(cls, obj):
"""
Takes an object extracted by the json parser and decodes the
special-formating dictionaries used to store special types.
"""
if isinstance(obj, dict) and 'data' in obj:
if len(obj) == 2 and '__ComplexList__' in obj:
return [complex(*v) for v in obj['data']]
elif len(obj) == 2 and '__QQList__' in obj:
return [QQ(tuple(v)) for v in obj['data']]
elif len(obj) == 3 and '__NFList__' in obj and 'base' in obj:
base = cls.extract(obj['base'])
return [cls._extract(base, c) for c in obj['data']]
elif len(obj) == 2 and '__IntDict__' in obj:
return {Integer(k): cls.extract(v) for k,v in obj['data']}
elif len(obj) == 3 and '__Vector__' in obj and 'base' in obj:
base = cls.extract(obj['base'])
return vector([cls._extract(base, v) for v in obj['data']])
elif len(obj) == 2 and '__Rational__' in obj:
return Rational(*obj['data'])
elif len(obj) == 3 and '__RealLiteral__' in obj and 'prec' in obj:
return LmfdbRealLiteral(RealField(obj['prec']), obj['data'])
elif len(obj) == 2 and '__complex__' in obj:
return complex(*obj['data'])
elif len(obj) == 3 and '__Complex__' in obj and 'prec' in obj:
return ComplexNumber(ComplexField(obj['prec']), *obj['data'])
elif len(obj) == 3 and '__NFElt__' in obj and 'parent' in obj:
return cls._extract(cls.extract(obj['parent']), obj['data'])
elif len(obj) == 3 and ('__NFRelative__' in obj or '__NFAbsolute__' in obj) and 'vname' in obj:
poly = cls.extract(obj['data'])
return NumberField(poly, name=obj['vname'])
elif len(obj) == 2 and '__NFCyclotomic__' in obj:
return CyclotomicField(obj['data'])
elif len(obj) == 2 and '__IntegerRing__' in obj:
return ZZ
elif len(obj) == 2 and '__RationalField__' in obj:
return QQ
elif len(obj) == 3 and '__RationalPoly__' in obj and 'vname' in obj:
return QQ[obj['vname']]([QQ(tuple(v)) for v in obj['data']])
elif len(obj) == 4 and '__Poly__' in obj and 'vname' in obj and 'base' in obj:
base = cls.extract(obj['base'])
return base[obj['vname']]([cls._extract(base, c) for c in obj['data']])
elif len(obj) == 5 and '__PowerSeries__' in obj and 'vname' in obj and 'base' in obj and 'prec' in obj:
base = cls.extract(obj['base'])
prec = infinity if obj['prec'] == 'inf' else int(obj['prec'])
return base[[obj['vname']]]([cls._extract(base, c) for c in obj['data']], prec=prec)
elif len(obj) == 2 and '__date__' in obj:
return datetime.datetime.strptime(obj['data'], "%Y-%m-%d").date()
elif len(obj) == 2 and '__time__' in obj:
return datetime.datetime.strptime(obj['data'], "%H:%M:%S.%f").time()
elif len(obj) == 2 and '__datetime__' in obj:
return datetime.datetime.strptime(obj['data'], "%Y-%m-%d %H:%M:%S.%f")
return obj
def sage_character_to_magma(chi,N=None,magma=None):
if magma is None:
magma = sage.interfaces.magma
if N is None:
N = chi.modulus()
else:
N = ZZ(N)
chi = chi.extend(N)
G = chi.parent()
order = chi.order()
gens = G.unit_gens()
ElGm = magma.DirichletGroupFull(N)
Kcyc = CyclotomicField(ElGm.BaseRing().CyclotomicOrder(),names='z')
phi = ElGm.BaseRing().sage().hom([Kcyc.gen()])
target = [chi(g) for g in gens]
for chim in ElGm.Elements():
if chim.Order().sage() == order:
this = [phi(chim.Evaluate(g).sage()) for g in gens]
if all((u == v for u, v in zip(this, target))):
return chim
raise RuntimeError("Should not get to this point")
def eisenstein_series_at_inf(phi, psi, k, prec=10, t=1, base_ring=None):
r"""
Return Fourier expansion of Eistenstein series at a cusp.
INPUT:
- ``phi`` -- Dirichlet character.
- ``psi`` -- Dirichlet character.
- ``k`` -- integer, the weight of the Eistenstein series.
- ``prec`` -- integer (default: 10).
- ``t`` -- integer (default: 1).
OUTPUT:
The Fourier expansion of the Eisenstein series $E_k^{\phi,\psi, t}$ (as
defined by [Diamond-Shurman]) at the specific cusp.
EXAMPLES:
sage: phi = DirichletGroup(3)[1]
sage: psi = DirichletGroup(5)[1]
sage: E = eisenstein_series_at_inf(phi, psi, 4)
"""
N1, N2 = phi.level(), psi.level()
N = N1 * N2
#The Fourier expansion of the Eisenstein series at infinity is in the field Q(zeta_Ncyc)
Ncyc = lcm([euler_phi(N1), euler_phi(N2)])
if base_ring == None:
base_ring = CyclotomicField(Ncyc)
Q = PowerSeriesRing(base_ring, 'q')
q = Q.gen()
s = O(q**prec)
#Weight 2 with trivial characters is calculated separately
if k == 2 and phi.conductor() == 1 and psi.conductor() == 1:
if t == 1:
raise TypeError('E_2 is not a modular form.')
s = 1 / 24 * (t - 1)
for m in srange(1, prec):
for n in srange(1, prec / m):
s += n * (q**(m * n) - t * q**(m * n * t))
return s + O(q**prec)
if psi.level() == 1 and k == 1:
s -= phi.bernoulli(k) / k
elif phi.level() == 1:
s -= psi.bernoulli(k) / k
for m in srange(1, prec / t):
for n in srange(1, prec / t / m + 1):
s += 2 * base_ring(phi(m)) * base_ring(
psi(n)) * n**(k - 1) * q**(m * n * t)
return s + O(q**prec)
def eis_E(cv, dv, N, k, Q=None, param_level=1, prec=10):
r"""
Computes the coefficient of the Eisenstein series for $\Gamma(N)$.
Not intended to be called by user.
INPUT:
- cv - int, the first coordinate of the vector determining the \Gamma(N)
Eisenstein series
- dv - int, the second coordinate of the vector determining the \Gamma(N)
Eisenstein series
- N - int, the level of the Eisenstein series to be computed
- k - int, the weight of the Eisenstein seriess to be computed
- Q - power series ring, the ring containing the q-expansion to be computed
- param_level - int, the parameter of the returned series will be
q_{param_level}
- prec - int, the precision. The series in q_{param_level} will be truncated
after prec coefficients
OUTPUT:
- an element of the ring Q, which is the Fourier expansion of the Eisenstein
series
"""
if Q == None:
Q = PowerSeriesRing(CyclotomicField(N), 'q')
R = Q.base_ring()
zetaN = R.zeta(N)
q = Q.gen()
cv = cv % N
dv = dv % N
#if dv == 0 and cv == 0 and k == 2:
# raise ValueError("E_2 is not a modular form")
if k == 1:
if cv == 0 and dv == 0:
raise ValueError("that shouldn't have happened...")
elif cv == 0 and dv != 0:
s = QQ(1) / QQ(2) * (1 + zetaN**dv) / (1 - zetaN**dv)
elif cv != 0:
s = QQ(1) / QQ(2) - QQ(cv) / QQ(N) + floor(QQ(cv) / QQ(N))
elif k > 1:
if cv == 0:
s = hurwitz_hat(QQ(dv), QQ(N), 1 - k, zetaN)
else:
s = 0
for n1 in xrange(1, prec): # this is n/m in DS
for n2 in xrange(1, prec / n1 + 1): # this is m in DS
if Mod(n1, N) == Mod(cv, N):
s += n2**(k - 1) * zetaN**(dv * n2) * q**(n1 * n2)
if Mod(n1, N) == Mod(-cv, N):
s += (-1)**k * n2**(k - 1) * zetaN**(-dv * n2) * q**(n1 * n2)
return s + O(q**floor(prec))
def field(self):
r"""
Return a cyclotomic field large enough to
contain the `2 \ell`-th roots of unity, as well as
all the S-matrix entries.
EXAMPLES::
sage: FusionRing("A2",2).field()
Cyclotomic Field of order 60 and degree 16
sage: FusionRing("B2",2).field()
Cyclotomic Field of order 40 and degree 16
"""
return CyclotomicField(4 * self._cyclotomic_order)
def eis_F(cv, dv, N, k, Q=None, prec=10, t=1):
"""
Computes the coefficient of the Eisenstein series for $\Gamma(N)$.
Not indented to be called by user.
INPUT:
- cv - int, the first coordinate of the vector determining the \Gamma(N)
Eisenstein series
- dv - int, the second coordinate of the vector determining the \Gamma(N)
Eisenstein series
- N - int, the level of the Eisenstein series to be computed
- k - int, the weight of the Eisenstein seriess to be computed
- Q - power series ring, the ring containing the q-expansion to be computed
- param_level - int, the parameter of the returned series will be
q_{param_level}
- prec - int, the precision. The series in q_{param_level} will be truncated
after prec coefficients
OUTPUT:
- an element of the ring Q, which is the Fourier expansion of the Eisenstein
series
"""
if Q == None:
Q = PowerSeriesRing(CyclotomicField(N), 'q{}'.format(N))
R = Q.base_ring()
zetaN = R.zeta(N)
q = Q.gen()
s = 0
if k == 1:
if cv % N == 0 and dv % N != 0:
s = QQ(1) / QQ(2) * (1 + zetaN**dv) / (1 - zetaN**dv)
elif cv % N != 0:
s = QQ(1) / QQ(2) - QQ(cv) / QQ(N) + floor(QQ(cv) / QQ(N))
elif k > 1:
s = -ber_pol(QQ(cv) / QQ(N) - floor(QQ(cv) / QQ(N)), k) / QQ(k)
for n1 in xrange(1, ceil(prec / QQ(t))): # this is n/m in DS
for n2 in xrange(1,
ceil(prec / QQ(t) / QQ(n1)) + 1): # this is m in DS
if Mod(n1, N) == Mod(cv, N):
s += N**(1 - k) * n1**(k - 1) * zetaN**(dv * n2) * q**(t * n1 *
n2)
if Mod(n1, N) == Mod(-cv, N):
s += (-1)**k * N**(1 - k) * n1**(k - 1) * zetaN**(
-dv * n2) * q**(t * n1 * n2)
return s + O(q**floor(prec))
def toCyclPowerBase(M, order):
"""
Let's
K = CyclotomicField(order).
INPUT:
- `M` -- A matrix over the cyclomotic field `K`.
- `order` -- The order of `K`, the cyclomotic field.
OUTPUT:
- A list of matrices `ms` in power base where every matrix
is a factor to `zeta**i` where `zeta = K.gen()`
and `len(ms) == K.degree()`.
"""
K = CyclotomicField(order)
zeta = K.gen()
Kcoords = zeta.coordinates_in_terms_of_powers()
assert len(K.power_basis()) == K.degree()
ms = [matrix(QQ,M.nrows(),M.ncols()) for i in range(K.degree())]
for y in range(M.nrows()):
for x in range(M.ncols()):
try:
v_ = M[y,x]
v = K(v_)
coords = Kcoords(v)
except TypeError:
print "type of {1} ({2}) is not valid in Cyclomotic field of order {0}".format(order, M[y,x], type(M[y,x]))
raise
assert len(coords) == K.degree()
for i in range(K.degree()):
ms[i][y,x] = coords[i]
return ms
def _element_constructor_(self, elt):
r"""
TESTS::
sage: UCF = UniversalCyclotomicField()
sage: UCF(3)
3
sage: UCF(3/2)
3/2
sage: C = CyclotomicField(13)
sage: UCF(C.gen())
E(13)
sage: UCF(C.gen() - 3*C.gen()**2 + 5*C.gen()**5)
E(13) - 3*E(13)^2 + 5*E(13)^5
sage: C = CyclotomicField(12)
sage: zeta12 = C.gen()
sage: a = UCF(zeta12 - 3* zeta12**2)
sage: a
-E(12)^7 + 3*E(12)^8
sage: C(_) == a
True
sage: UCF('[[0, 1], [0, 2]]')
Traceback (most recent call last):
...
TypeError: [ [ 0, 1 ], [ 0, 2 ] ] of type <type
'sage.libs.gap.element.GapElement_List'> not valid to initialize an
element of the universal cyclotomic field
.. TODO::
Implement conversion from QQbar (and as a consequence from the
symbolic ring)
"""
elt = py_scalar_to_element(elt)
if isinstance(elt, (Integer, Rational)):
return self.element_class(self, libgap(elt))
elif isinstance(elt, (GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)):
return self.element_class(self, elt)
elif not elt:
return self.zero()
obj = None
if isinstance(elt, gap.GapElement):
obj = libgap(elt)
elif isinstance(elt, gap3.GAP3Element):
obj = libgap.eval(str(elt))
elif isinstance(elt, str):
obj = libgap.eval(elt)
if obj is not None:
if not isinstance(obj, (GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)):
raise TypeError("{} of type {} not valid to initialize an element of the universal cyclotomic field".format(obj, type(obj)))
return self.element_class(self, obj)
# late import to avoid slowing down the above conversions
from sage.rings.number_field.number_field_element import NumberFieldElement
from sage.rings.number_field.number_field import NumberField_cyclotomic, CyclotomicField
P = parent(elt)
if isinstance(elt, NumberFieldElement) and isinstance(P, NumberField_cyclotomic):
n = P.gen().multiplicative_order()
elt = CyclotomicField(n)(elt)
return sum(c * self.gen(n, i)
for i, c in enumerate(elt._coefficients()))
else:
raise TypeError("{} of type {} not valid to initialize an element of the universal cyclotomic field".format(elt, type(elt)))
def attack(m, q, r=4, sigma=3.0, subfield_only=False):
K = CyclotomicField(m, 'z')
z = K.gen()
OK = K.ring_of_integers()
G = K.galois_group()
n = euler_phi(m)
mprime = m / r
nprime = euler_phi(mprime)
Gprime = [tau for tau in G if tau(z**r) == z**r]
R = PolynomialRing(IntegerRing(), 'a')
a = R.gen()
phim = a**n + 1
D = DiscreteGaussianDistributionIntegerSampler(sigma)
print "sampling f,g"
while True:
f = sum([D() * z**i for i in range(n)])
fx = sum([f[i] * a**i for i in range(n)])
res = inverse(fx, phim, q)
if res[0]:
f_inv = sum([res[1][i] * z**i for i in range(n)])
print "f_inv * f = %s (mod %d)" % ((f * f_inv).mod(q), q)
break
g = sum([D() * z**i for i in range(n)])
print "done sampling f, g"
#h = [g*f^{-1)]_q
h = (g * f_inv).mod(q)
lognorm_f = log(f.vector().norm(), 2)
lognorm_g = log(g.vector().norm(), 2)
print "f*h - g = %s" % (f * h - g).mod(q)
print "log q = ", log(q, 2).n(precision)
print "log |f| = %s, log |g| = %s" % (lognorm_f.n(precision),
lognorm_g.n(precision))
print "log |(f,g)| = ", log(
sqrt(f.vector().norm()**2 + g.vector().norm()**2), 2).n(precision)
print "begin computing N(f), N(g), N(h), Tr(h), fbar"
fprime = norm(f, Gprime)
gprime = norm(g, Gprime)
hprime = norm(h, Gprime).mod(q)
htr = trace(h, Gprime)
fbar = prod([tau(f) for tau in Gprime[1:]])
print "end computing N(f), N(g), N(h), Tr(h), fbar"
lognorm_fp = log(fprime.vector().norm(), 2)
lognorm_gp = log(gprime.vector().norm(), 2)
print "%d * log |f| - log |f'| = %s" % (r, r * lognorm_f.n(precision) -
lognorm_fp.n(precision))
print "log |(f', g')| = ", log(
sqrt(fprime.vector().norm()**2 + gprime.vector().norm()**2),
2).n(precision)
print "log |N(f), Tr(g fbar)| = ", log(
sqrt(fprime.vector().norm()**2 +
trace(g * fbar, Gprime).vector().norm()**2), 2).n(precision)
#(fprime, gprime) lies in the lattice \Lambda_hprime^q
print "f'*h' - g' = %s " % (hprime * fprime - gprime).mod(q)
print "N(f) Tr(h) - Tr(g fbar) = %s" % (htr * fprime -
trace(g * fbar, Gprime)).mod(q)
if not subfield_only:
ntru_full = NTRU(h, K, q)
full_sv = ntru_full.shortest_vector()
print "log |v| = %s" % log(full_sv.norm(), 2).n(precision)
ntru_subfield = NTRU_subfield(hprime, q, nprime, r)
ntru_trace_subfield = NTRU_subfield(htr, q, nprime, r)
print "begin computing Shortest Vector of subfield lattice"
norm_sv = ntru_subfield.shortest_vector()
tr_sv = ntru_trace_subfield.shortest_vector()
print "end computing Shortest Vector of subfield lattice"
norm_xp = sum(
[coerce(Integer, norm_sv[i]) * z**(r * i) for i in range(nprime)])
tr_xp = sum(
[coerce(Integer, tr_sv[i]) * z**(r * i) for i in range(nprime)])
print "Norm map: log |(x',y')| = ", log(norm_sv.norm(), 2).n(precision)
print "Trace map: log |(x', y')| = ", log(tr_sv.norm(), 2).n(precision)
#test if xprime belongs to <fprime>
mat = []
for i in range(nprime):
coordinate = (fprime * z**(r * i)).vector().list()
mat.append([coordinate[r * j] for j in range(nprime)])
FL = IntegerLattice(mat)
print norm_sv[:nprime] in FL
print tr_sv[:nprime] in FL
norm_x = norm_xp
norm_y = mod_q(norm_x * h, q)
tr_x = tr_xp
tr_y = mod_q(tr_x * h, q)
print "Norm map: log |(x,y)| = ", log(
sqrt(norm_x.vector().norm()**2 + norm_y.vector().norm()**2),
2).n(precision)
print "Trace map: log |(x,y)| = ", log(
sqrt(tr_x.vector().norm()**2 + tr_y.vector().norm()**2),
2).n(precision)
def product_space(chi, k, weights=False, base_ring=None, verbose=False):
r"""
Computes all eisenstein series, and products of pairs of eisenstein series
of lower weight, lying in the space of modular forms of weight $k$ and
nebentypus $\chi$.
INPUT:
- chi - Dirichlet character, the nebentypus of the target space
- k - an integer, the weight of the target space
OUTPUT:
- a matrix of coefficients of q-expansions, which are the products of
Eisenstein series in M_k(chi).
WARNING: It is only for principal chi that we know that the resulting
space is the whole space of modular forms.
"""
if weights == False:
weights = srange(1, k / 2 + 1)
weight_dict = {}
weight_dict[-1] = [w for w in weights if w % 2] # Odd weights
weight_dict[1] = [w for w in weights if not w % 2] # Even weights
try:
N = chi.modulus()
except AttributeError:
if chi.parent() == ZZ:
N = chi
chi = DirichletGroup(N)[0]
Id = DirichletGroup(1)[0]
if chi(-1) != (-1)**k:
raise ValueError('chi(-1)!=(-1)^k')
sturm = ModularForms(N, k).sturm_bound() + 1
if N > 1:
target_dim = dimension_modular_forms(chi, k)
else:
target_dim = dimension_modular_forms(1, k)
D = DirichletGroup(N)
# product_space should ideally be called over number fields. Over complex
# numbers the exact linear algebra solutions might not exist.
if base_ring == None:
base_ring = CyclotomicField(euler_phi(N))
Q = PowerSeriesRing(base_ring, 'q')
q = Q.gen()
d = len(D)
prim_chars = [phi.primitive_character() for phi in D]
divs = divisors(N)
products = Matrix(base_ring, [])
indexlist = []
rank = 0
if verbose:
print(D)
print('Sturm bound', sturm)
#TODO: target_dim needs refinment in the case of weight 2.
print('Target dimension', target_dim)
for i in srange(0, d): # First character
phi = prim_chars[i]
M1 = phi.conductor()
for j in srange(0, d): # Second character
psi = prim_chars[j]
M2 = psi.conductor()
if not M1 * M2 in divs:
continue
parity = psi(-1) * phi(-1)
for t1 in divs:
if not M1 * M2 * t1 in divs:
continue
#TODO: THE NEXT CONDITION NEEDS TO BE CORRECTED. THIS IS JUST A TEST
if phi.bar() == psi and not (
k == 2): #and i==0 and j==0 and t1==1):
E = eisenstein_series_at_inf(phi, psi, k, sturm, t1,
base_ring).padded_list()
try:
products.T.solve_right(vector(base_ring, E))
except ValueError:
products = Matrix(products.rows() + [E])
indexlist.append([k, i, j, t1])
rank += 1
if verbose:
print('Added ', [k, i, j, t1])
print('Rank is now', rank)
if rank == target_dim:
return products, indexlist
for t in divs:
if not M1 * M2 * t1 * t in divs:
continue
for t2 in divs:
if not M1 * M2 * t1 * t2 * t in divs:
continue
for l in weight_dict[parity]:
if l == 1 and phi.is_odd():
continue
if i == 0 and j == 0 and (l == 2 or l == k - 2):
continue
#TODO: THE NEXT CONDITION NEEDS TO BE REMOVED. THIS IS JUST A TEST
if l == 2 or l == k - 2:
continue
E1 = eisenstein_series_at_inf(
phi, psi, l, sturm, t1 * t, base_ring)
E2 = eisenstein_series_at_inf(
phi**(-1), psi**(-1), k - l, sturm, t2 * t,
base_ring)
#If chi is non-principal this needs to be changed to be something like chi*phi^(-1) instead of phi^(-1)
E = (E1 * E2 + O(q**sturm)).padded_list()
try:
products.T.solve_right(vector(base_ring, E))
except ValueError:
products = Matrix(products.rows() + [E])
indexlist.append([l, k - l, i, j, t1, t2, t])
rank += 1
if verbose:
print('Added ',
[l, k - l, i, j, t1, t2, t])
print('Rank', rank)
if rank == target_dim:
return products, indexlist
return products, indexlist
def _element_constructor_(self, elt):
r"""
TESTS::
sage: UCF = UniversalCyclotomicField()
sage: UCF(3)
3
sage: UCF(3/2)
3/2
sage: C = CyclotomicField(13)
sage: UCF(C.gen())
E(13)
sage: UCF(C.gen() - 3*C.gen()**2 + 5*C.gen()**5)
E(13) - 3*E(13)^2 + 5*E(13)^5
sage: C = CyclotomicField(12)
sage: zeta12 = C.gen()
sage: a = UCF(zeta12 - 3* zeta12**2)
sage: a
-E(12)^7 + 3*E(12)^8
sage: C(_) == a
True
sage: UCF('[[0, 1], [0, 2]]')
Traceback (most recent call last):
...
TypeError: [ [ 0, 1 ], [ 0, 2 ] ] of type <type
'sage.libs.gap.element.GapElement_List'> not valid to initialize an
element of the universal cyclotomic field
.. TODO::
Implement conversion from QQbar (and as a consequence from the
symbolic ring)
"""
elt = py_scalar_to_element(elt)
if isinstance(elt, (Integer, Rational)):
return self.element_class(self, libgap(elt))
elif isinstance(elt, (GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)):
return self.element_class(self, elt)
elif not elt:
return self.zero()
obj = None
if isinstance(elt, gap.GapElement):
obj = libgap(elt)
elif isinstance(elt, gap3.GAP3Element):
obj = libgap.eval(str(elt))
elif isinstance(elt, str):
obj = libgap.eval(elt)
if obj is not None:
if not isinstance(obj, (GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)):
raise TypeError("{} of type {} not valid to initialize an element of the universal cyclotomic field".format(obj, type(obj)))
return self.element_class(self, obj)
# late import to avoid slowing down the above conversions
from sage.rings.number_field.number_field_element import NumberFieldElement
from sage.rings.number_field.number_field import NumberField_cyclotomic, CyclotomicField
P = parent(elt)
if isinstance(elt, NumberFieldElement) and isinstance(P, NumberField_cyclotomic):
n = P.gen().multiplicative_order()
elt = CyclotomicField(n)(elt)
coeffs = elt._coefficients()
return sum(c * self.gen(n,i) for i,c in enumerate(elt._coefficients()))
else:
raise TypeError("{} of type {} not valid to initialize an element of the universal cyclotomic field".format(elt, type(elt)))
def dft(self, chi = lambda x: x):
"""
A discrete Fourier transform "over `\QQ`" using exact
`N`-th roots of unity.
EXAMPLES::
sage: J = range(6)
sage: A = [ZZ(1) for i in J]
sage: s = IndexedSequence(A,J)
sage: s.dft(lambda x:x^2)
Indexed sequence: [6, 0, 0, 6, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: s.dft()
Indexed sequence: [6, 0, 0, 0, 0, 0]
indexed by [0, 1, 2, 3, 4, 5]
sage: G = SymmetricGroup(3)
sage: J = G.conjugacy_classes_representatives()
sage: s = IndexedSequence([1,2,3],J) # 1,2,3 are the values of a class fcn on G
sage: s.dft() # the "scalar-valued Fourier transform" of this class fcn
Indexed sequence: [8, 2, 2]
indexed by [(), (1,2), (1,2,3)]
sage: J = AbelianGroup(2,[2,3],names='ab')
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Multiplicative Abelian group isomorphic to C2 x C3
sage: J = CyclicPermutationGroup(6)
sage: s = IndexedSequence([1,2,3,4,5,6],J)
sage: s.dft() # the precision of output is somewhat random and architecture dependent.
Indexed sequence: [21.0000000000000, -2.99999999999997 - 1.73205080756885*I, -2.99999999999999 + 1.73205080756888*I, -9.00000000000000 + 0.0000000000000485744257349999*I, -0.00000000000000976996261670137 - 0.0000000000000159872115546022*I, -0.00000000000000621724893790087 - 0.0000000000000106581410364015*I]
indexed by Cyclic group of order 6 as a permutation group
sage: p = 7; J = range(p); A = [kronecker_symbol(j,p) for j in J]
sage: s = IndexedSequence(A,J)
sage: Fs = s.dft()
sage: c = Fs.list()[1]; [x/c for x in Fs.list()]; s.list()
[0, 1, 1, -1, 1, -1, -1]
[0, 1, 1, -1, 1, -1, -1]
The DFT of the values of the quadratic residue symbol is itself, up to
a constant factor (denoted c on the last line above).
.. TODO::
Read the parent of the elements of S; if `\QQ` or `\CC` leave as
is; if AbelianGroup, use abelian_group_dual; if some other
implemented Group (permutation, matrix), call .characters()
and test if the index list is the set of conjugacy classes.
"""
J = self.index_object() ## index set of length N
N = len(J)
S = self.list()
F = self.base_ring() ## elements must be coercible into QQ(zeta_N)
if not(J[0] in ZZ):
G = J[0].parent() ## if J is not a range it is a group G
if J[0] in ZZ and F.base_ring().fraction_field()==QQ:
## assumes J is range(N)
zeta = CyclotomicField(N).gen()
FT = [sum([S[i]*chi(zeta**(i*j)) for i in J]) for j in J]
elif not(J[0] in ZZ) and G.is_abelian() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
if is_PermutationGroupElement(J[0]):
## J is a CyclicPermGp
n = G.order()
a = list(factor(n))
invs = [x[0]**x[1] for x in a]
G = AbelianGroup(len(a),invs)
## assumes J is AbelianGroup(...)
Gd = G.dual_group()
FT = [sum([S[i]*chid(G.list()[i]) for i in range(N)])
for chid in Gd]
elif not(J[0] in ZZ) and G.is_finite() and F == ZZ or (F.is_field() and F.base_ring()==QQ):
## assumes J is the list of conj class representatives of a
## PermuationGroup(...) or Matrixgroup(...)
chi = G.character_table()
FT = [sum([S[i]*chi[i,j] for i in range(N)]) for j in range(N)]
else:
raise ValueError("list elements must be in QQ(zeta_"+str(N)+")")
return IndexedSequence(FT,J)
def molien_series(self, chi=None, return_series=True, prec=20, variable='t'):
r"""
Compute the Molien series of this finite group with respect to the
character ``chi``. It can be returned either as a rational function
in one variable or a power series in one variable. The base field
must be a finite field, the rationals, or a cyclotomic field.
Note that the base field characteristic cannot divide the group
order (i.e., the non-modular case).
ALGORITHM:
For a finite group `G` in characteristic zero we construct the Molien series as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\text{det}(I-tg)},
where `I` is the identity matrix and `t` an indeterminate.
For characteristic `p` not dividing the order of `G`, let `k` be the base field
and `N` the order of `G`. Define `\lambda` as a primitive `N`-th root of unity over `k`
and `\omega` as a primitive `N`-th root of unity over `\QQ`. For each `g \in G`
define `k_i(g)` to be the positive integer such that
`e_i = \lambda^{k_i(g)}` for each eigenvalue `e_i` of `g`. Then the Molien series
is computed as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\prod_{i=1}^n(1 - t\omega^{k_i(g)})},
where `t` is an indeterminant. [Dec1998]_
INPUT:
- ``chi`` -- (default: trivial character) a linear group character of this group
- ``return_series`` -- boolean (default: ``True``) if ``True``, then returns
the Molien series as a power series, ``False`` as a rational function
- ``prec`` -- integer (default: 20); power series default precision
- ``variable`` -- string (default: ``'t'``); Variable name for the Molien series
OUTPUT: single variable rational function or power series with integer coefficients
EXAMPLES::
sage: MatrixGroup(matrix(QQ,2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups
sage: MatrixGroup(matrix(GF(3),2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: characteristic cannot divide group order
Tetrahedral Group::
sage: K.<i> = CyclotomicField(4)
sage: Tetra = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [0,i, -i,0])
sage: Tetra.molien_series(prec=30)
1 + t^8 + 2*t^12 + t^16 + 2*t^20 + 3*t^24 + 2*t^28 + O(t^30)
sage: mol = Tetra.molien_series(return_series=False); mol
(t^8 - t^4 + 1)/(t^16 - t^12 - t^4 + 1)
sage: mol.parent()
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
sage: chi = Tetra.character(Tetra.character_table()[1])
sage: Tetra.molien_series(chi, prec=30, variable='u')
u^6 + u^14 + 2*u^18 + u^22 + 2*u^26 + 3*u^30 + 2*u^34 + O(u^36)
sage: chi = Tetra.character(Tetra.character_table()[2])
sage: Tetra.molien_series(chi)
t^10 + t^14 + t^18 + 2*t^22 + 2*t^26 + O(t^30)
::
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: mol = S3.molien_series(prec=10); mol
1 + t + 2*t^2 + 3*t^3 + 4*t^4 + 5*t^5 + 7*t^6 + 8*t^7 + 10*t^8 + 12*t^9 + O(t^10)
sage: mol.parent()
Power Series Ring in t over Integer Ring
Octahedral Group::
sage: K.<v> = CyclotomicField(8)
sage: a = v-v^3 #sqrt(2)
sage: i = v^2
sage: Octa = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [(1+i)/a,0, 0,(1-i)/a])
sage: Octa.molien_series(prec=30)
1 + t^8 + t^12 + t^16 + t^18 + t^20 + 2*t^24 + t^26 + t^28 + O(t^30)
Icosahedral Group::
sage: K.<v> = CyclotomicField(10)
sage: z5 = v^2
sage: i = z5^5
sage: a = 2*z5^3 + 2*z5^2 + 1 #sqrt(5)
sage: Ico = MatrixGroup([[z5^3,0, 0,z5^2], [0,1, -1,0], [(z5^4-z5)/a, (z5^2-z5^3)/a, (z5^2-z5^3)/a, -(z5^4-z5)/a]])
sage: Ico.molien_series(prec=40)
1 + t^12 + t^20 + t^24 + t^30 + t^32 + t^36 + O(t^40)
::
sage: G = MatrixGroup(CyclicPermutationGroup(3))
sage: chi = G.character(G.character_table()[1])
sage: G.molien_series(chi, prec=10)
t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 9*t^6 + 12*t^7 + 15*t^8 + 18*t^9 + 22*t^10 + O(t^11)
::
sage: K = GF(5)
sage: S = MatrixGroup(SymmetricGroup(4))
sage: G = MatrixGroup([matrix(K,4,4,[K(y) for u in m.list() for y in u])for m in S.gens()])
sage: G.molien_series(return_series=False)
1/(t^10 - t^9 - t^8 + 2*t^5 - t^2 - t + 1)
::
sage: i = GF(7)(3)
sage: G = MatrixGroup([[i^3,0,0,-i^3],[i^2,0,0,-i^2]])
sage: chi = G.character(G.character_table()[4])
sage: G.molien_series(chi)
3*t^5 + 6*t^11 + 9*t^17 + 12*t^23 + O(t^25)
"""
if not self.is_finite():
raise NotImplementedError("only implemented for finite groups")
if chi is None:
chi = self.trivial_character()
M = self.matrix_space()
R = FractionField(self.base_ring())
N = self.order()
if R.characteristic() == 0:
P = PolynomialRing(R, variable)
t = P.gen()
#it is possible the character is over a larger cyclotomic field
K = chi.values()[0].parent()
if K.degree() != 1:
if R.degree() != 1:
L = K.composite_fields(R)[0]
else:
L = K
else:
L = R
mol = P(0)
for g in self:
mol += L(chi(g)) / (M.identity_matrix()-t*g.matrix()).det().change_ring(L)
elif R.characteristic().divides(N):
raise NotImplementedError("characteristic cannot divide group order")
else: #char p>0
#find primitive Nth roots of unity over base ring and QQ
F = cyclotomic_polynomial(N).change_ring(R)
w = F.roots(ring=R.algebraic_closure(), multiplicities=False)[0]
#don't need to extend further in this case since the order of
#the roots of unity in the character divide the order of the group
L = CyclotomicField(N, 'v')
v = L.gen()
#construct Molien series
P = PolynomialRing(L, variable)
t = P.gen()
mol = P(0)
for g in self:
#construct Phi
phi = L(chi(g))
for e in g.matrix().eigenvalues():
#find power such that w**n = e
n = 1
while w**n != e and n < N+1:
n += 1
#raise v to that power
phi *= (1-t*v**n)
mol += P(1)/phi
#We know the coefficients will be integers
mol = mol.numerator().change_ring(ZZ) / mol.denominator().change_ring(ZZ)
#divide by group order
mol /= N
if return_series:
PS = PowerSeriesRing(ZZ, variable, default_prec=prec)
return PS(mol)
return mol
def Omega_ge(a, exponents):
r"""
Return `\Omega_{\ge}` of the expression specified by the input.
To be more precise, calculate
.. MATH::
\Omega_{\ge} \frac{\mu^a}{
(1 - z_0 \mu^{e_0}) \dots (1 - z_{n-1} \mu^{e_{n-1}})}
and return its numerator and a factorization of its denominator.
Note that `z_0`, ..., `z_{n-1}` only appear in the output, but not in the
input.
INPUT:
- ``a`` -- an integer
- ``exponents`` -- a tuple of integers
OUTPUT:
A pair representing a quotient as follows: Its first component is the
numerator as a Laurent polynomial, its second component a factorization
of the denominator as a tuple of Laurent polynomials, where each
Laurent polynomial `z` represents a factor `1 - z`.
The parents of these Laurent polynomials is always a
Laurent polynomial ring in `z_0`, ..., `z_{n-1}` over `\ZZ`, where
`n` is the length of ``exponents``.
EXAMPLES::
sage: from sage.rings.polynomial.omega import Omega_ge
sage: Omega_ge(0, (1, -2))
(1, (z0, z0^2*z1))
sage: Omega_ge(0, (1, -3))
(1, (z0, z0^3*z1))
sage: Omega_ge(0, (1, -4))
(1, (z0, z0^4*z1))
sage: Omega_ge(0, (2, -1))
(z0*z1 + 1, (z0, z0*z1^2))
sage: Omega_ge(0, (3, -1))
(z0*z1^2 + z0*z1 + 1, (z0, z0*z1^3))
sage: Omega_ge(0, (4, -1))
(z0*z1^3 + z0*z1^2 + z0*z1 + 1, (z0, z0*z1^4))
sage: Omega_ge(0, (1, 1, -2))
(-z0^2*z1*z2 - z0*z1^2*z2 + z0*z1*z2 + 1, (z0, z1, z0^2*z2, z1^2*z2))
sage: Omega_ge(0, (2, -1, -1))
(z0*z1*z2 + z0*z1 + z0*z2 + 1, (z0, z0*z1^2, z0*z2^2))
sage: Omega_ge(0, (2, 1, -1))
(-z0*z1*z2^2 - z0*z1*z2 + z0*z2 + 1, (z0, z1, z0*z2^2, z1*z2))
::
sage: Omega_ge(0, (2, -2))
(-z0*z1 + 1, (z0, z0*z1, z0*z1))
sage: Omega_ge(0, (2, -3))
(z0^2*z1 + 1, (z0, z0^3*z1^2))
sage: Omega_ge(0, (3, 1, -3))
(-z0^3*z1^3*z2^3 + 2*z0^2*z1^3*z2^2 - z0*z1^3*z2
+ z0^2*z2^2 - 2*z0*z2 + 1,
(z0, z1, z0*z2, z0*z2, z0*z2, z1^3*z2))
::
sage: Omega_ge(0, (3, 6, -1))
(-z0*z1*z2^8 - z0*z1*z2^7 - z0*z1*z2^6 - z0*z1*z2^5 - z0*z1*z2^4 +
z1*z2^5 - z0*z1*z2^3 + z1*z2^4 - z0*z1*z2^2 + z1*z2^3 -
z0*z1*z2 + z0*z2^2 + z1*z2^2 + z0*z2 + z1*z2 + 1,
(z0, z1, z0*z2^3, z1*z2^6))
TESTS::
sage: Omega_ge(0, (2, 2, 1, 1, 1, -1, -1))[0].number_of_terms() # long time
1695
sage: Omega_ge(0, (2, 2, 1, 1, 1, 1, 1, -1, -1))[0].number_of_terms() # not tested (too long, 1 min)
27837
::
sage: Omega_ge(1, (2,))
(1, (z0,))
"""
import logging
logger = logging.getLogger(__name__)
logger.info('Omega_ge: a=%s, exponents=%s', a, exponents)
from sage.arith.all import lcm, srange
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
from sage.rings.number_field.number_field import CyclotomicField
if not exponents or any(e == 0 for e in exponents):
raise NotImplementedError
rou = sorted(set(abs(e) for e in exponents) - set([1]))
ellcm = lcm(rou)
B = CyclotomicField(ellcm, 'zeta')
zeta = B.gen()
z_names = tuple('z{}'.format(i) for i in range(len(exponents)))
L = LaurentPolynomialRing(B, ('t', ) + z_names, len(z_names) + 1)
t = L.gens()[0]
Z = LaurentPolynomialRing(ZZ, z_names, len(z_names))
powers = {i: L(zeta**(ellcm // i)) for i in rou}
powers[2] = L(-1)
powers[1] = L(1)
exponents_and_values = tuple(
(e, tuple(powers[abs(e)]**j * z for j in srange(abs(e))))
for z, e in zip(L.gens()[1:], exponents))
x = tuple(v for e, v in exponents_and_values if e > 0)
y = tuple(v for e, v in exponents_and_values if e < 0)
def subs_power(expression, var, exponent):
r"""
Substitute ``var^exponent`` by ``var`` in ``expression``.
It is assumed that ``var`` only occurs with exponents
divisible by ``exponent``.
"""
p = tuple(var.dict().popitem()[0]).index(
1) # var is the p-th generator
def subs_e(e):
e = list(e)
assert e[p] % exponent == 0
e[p] = e[p] // exponent
return tuple(e)
parent = expression.parent()
result = parent(
{subs_e(e): c
for e, c in iteritems(expression.dict())})
return result
def de_power(expression):
expression = Z(expression)
for e, var in zip(exponents, Z.gens()):
if abs(e) == 1:
continue
expression = subs_power(expression, var, abs(e))
return expression
logger.debug('Omega_ge: preparing denominator')
factors_denominator = tuple(
de_power(1 - factor) for factor in _Omega_factors_denominator_(x, y))
logger.debug('Omega_ge: preparing numerator')
numerator = de_power(_Omega_numerator_(a, x, y, t))
logger.info('Omega_ge: completed')
return numerator, factors_denominator
def Omega_ge(a, exponents):
r"""
Return `\Omega_{\ge}` of the expression specified by the input.
To be more precise, calculate
.. MATH::
\Omega_{\ge} \frac{\mu^a}{
(1 - z_0 \mu^{e_0}) \dots (1 - z_{n-1} \mu^{e_{n-1}})}
and return its numerator and a factorization of its denominator.
Note that `z_0`, ..., `z_{n-1}` only appear in the output, but not in the
input.
INPUT:
- ``a`` -- an integer
- ``exponents`` -- a tuple of integers
OUTPUT:
A pair representing a quotient as follows: Its first component is the
numerator as a Laurent polynomial, its second component a factorization
of the denominator as a tuple of Laurent polynomials, where each
Laurent polynomial `z` represents a factor `1 - z`.
The parents of these Laurent polynomials is always a
Laurent polynomial ring in `z_0`, ..., `z_{n-1}` over `\ZZ`, where
`n` is the length of ``exponents``.
EXAMPLES::
sage: from sage.rings.polynomial.omega import Omega_ge
sage: Omega_ge(0, (1, -2))
(1, (z0, z0^2*z1))
sage: Omega_ge(0, (1, -3))
(1, (z0, z0^3*z1))
sage: Omega_ge(0, (1, -4))
(1, (z0, z0^4*z1))
sage: Omega_ge(0, (2, -1))
(z0*z1 + 1, (z0, z0*z1^2))
sage: Omega_ge(0, (3, -1))
(z0*z1^2 + z0*z1 + 1, (z0, z0*z1^3))
sage: Omega_ge(0, (4, -1))
(z0*z1^3 + z0*z1^2 + z0*z1 + 1, (z0, z0*z1^4))
sage: Omega_ge(0, (1, 1, -2))
(-z0^2*z1*z2 - z0*z1^2*z2 + z0*z1*z2 + 1, (z0, z1, z0^2*z2, z1^2*z2))
sage: Omega_ge(0, (2, -1, -1))
(z0*z1*z2 + z0*z1 + z0*z2 + 1, (z0, z0*z1^2, z0*z2^2))
sage: Omega_ge(0, (2, 1, -1))
(-z0*z1*z2^2 - z0*z1*z2 + z0*z2 + 1, (z0, z1, z0*z2^2, z1*z2))
::
sage: Omega_ge(0, (2, -2))
(-z0*z1 + 1, (z0, z0*z1, z0*z1))
sage: Omega_ge(0, (2, -3))
(z0^2*z1 + 1, (z0, z0^3*z1^2))
sage: Omega_ge(0, (3, 1, -3))
(-z0^3*z1^3*z2^3 + 2*z0^2*z1^3*z2^2 - z0*z1^3*z2
+ z0^2*z2^2 - 2*z0*z2 + 1,
(z0, z1, z0*z2, z0*z2, z0*z2, z1^3*z2))
::
sage: Omega_ge(0, (3, 6, -1))
(-z0*z1*z2^8 - z0*z1*z2^7 - z0*z1*z2^6 - z0*z1*z2^5 - z0*z1*z2^4 +
z1*z2^5 - z0*z1*z2^3 + z1*z2^4 - z0*z1*z2^2 + z1*z2^3 -
z0*z1*z2 + z0*z2^2 + z1*z2^2 + z0*z2 + z1*z2 + 1,
(z0, z1, z0*z2^3, z1*z2^6))
TESTS::
sage: Omega_ge(0, (2, 2, 1, 1, 1, 1, 1, -1, -1))[0].number_of_terms() # long time
27837
::
sage: Omega_ge(1, (2,))
(1, (z0,))
"""
import logging
logger = logging.getLogger(__name__)
logger.info('Omega_ge: a=%s, exponents=%s', a, exponents)
from sage.arith.all import lcm, srange
from sage.rings.integer_ring import ZZ
from sage.rings.polynomial.laurent_polynomial_ring import LaurentPolynomialRing
from sage.rings.number_field.number_field import CyclotomicField
if not exponents or any(e == 0 for e in exponents):
raise NotImplementedError
rou = sorted(set(abs(e) for e in exponents) - set([1]))
ellcm = lcm(rou)
B = CyclotomicField(ellcm, 'zeta')
zeta = B.gen()
z_names = tuple('z{}'.format(i) for i in range(len(exponents)))
L = LaurentPolynomialRing(B, ('t',) + z_names, len(z_names) + 1)
t = L.gens()[0]
Z = LaurentPolynomialRing(ZZ, z_names, len(z_names))
powers = {i: L(zeta**(ellcm//i)) for i in rou}
powers[2] = L(-1)
powers[1] = L(1)
exponents_and_values = tuple(
(e, tuple(powers[abs(e)]**j * z for j in srange(abs(e))))
for z, e in zip(L.gens()[1:], exponents))
x = tuple(v for e, v in exponents_and_values if e > 0)
y = tuple(v for e, v in exponents_and_values if e < 0)
def subs_power(expression, var, exponent):
r"""
Substitute ``var^exponent`` by ``var`` in ``expression``.
It is assumed that ``var`` only occurs with exponents
divisible by ``exponent``.
"""
p = tuple(var.dict().popitem()[0]).index(1) # var is the p-th generator
def subs_e(e):
e = list(e)
assert e[p] % exponent == 0
e[p] = e[p] // exponent
return tuple(e)
parent = expression.parent()
result = parent({subs_e(e): c for e, c in iteritems(expression.dict())})
return result
def de_power(expression):
expression = Z(expression)
for e, var in zip(exponents, Z.gens()):
if abs(e) == 1:
continue
expression = subs_power(expression, var, abs(e))
return expression
logger.debug('Omega_ge: preparing denominator')
factors_denominator = tuple(de_power(1 - factor)
for factor in _Omega_factors_denominator_(x, y))
logger.debug('Omega_ge: preparing numerator')
numerator = de_power(_Omega_numerator_(a, x, y, t))
logger.info('Omega_ge: completed')
return numerator, factors_denominator
def _element_constructor_(self, elt):
r"""
TESTS::
sage: UCF = UniversalCyclotomicField()
sage: UCF(3)
3
sage: UCF(3/2)
3/2
sage: C = CyclotomicField(13)
sage: UCF(C.gen())
E(13)
sage: UCF(C.gen() - 3*C.gen()**2 + 5*C.gen()**5)
E(13) - 3*E(13)^2 + 5*E(13)^5
sage: C = CyclotomicField(12)
sage: zeta12 = C.gen()
sage: a = UCF(zeta12 - 3* zeta12**2)
sage: a
-E(12)^7 + 3*E(12)^8
sage: C(_) == a
True
sage: UCF('[[0, 1], [0, 2]]')
Traceback (most recent call last):
...
TypeError: [ [ 0, 1 ], [ 0, 2 ] ]
of type <type 'sage.libs.gap.element.GapElement_List'> not valid
to initialize an element of the universal cyclotomic field
Some conversions from symbolic functions are possible::
sage: UCF = UniversalCyclotomicField()
sage: [UCF(sin(pi/k, hold=True)) for k in range(1,10)]
[0,
1,
-1/2*E(12)^7 + 1/2*E(12)^11,
1/2*E(8) - 1/2*E(8)^3,
-1/2*E(20)^13 + 1/2*E(20)^17,
1/2,
-1/2*E(28)^19 + 1/2*E(28)^23,
1/2*E(16)^3 - 1/2*E(16)^5,
-1/2*E(36)^25 + 1/2*E(36)^29]
sage: [UCF(cos(pi/k, hold=True)) for k in range(1,10)]
[-1,
0,
1/2,
1/2*E(8) - 1/2*E(8)^3,
-1/2*E(5)^2 - 1/2*E(5)^3,
-1/2*E(12)^7 + 1/2*E(12)^11,
-1/2*E(7)^3 - 1/2*E(7)^4,
1/2*E(16) - 1/2*E(16)^7,
-1/2*E(9)^4 - 1/2*E(9)^5]
.. TODO::
Implement conversion from QQbar (and as a consequence from the
symbolic ring)
"""
elt = py_scalar_to_element(elt)
if isinstance(elt, (Integer, Rational)):
return self.element_class(self, libgap(elt))
elif isinstance(
elt,
(GapElement_Integer, GapElement_Rational, GapElement_Cyclotomic)):
return self.element_class(self, elt)
elif not elt:
return self.zero()
obj = None
if isinstance(elt, gap.GapElement):
obj = libgap(elt)
elif isinstance(elt, gap3.GAP3Element):
obj = libgap.eval(str(elt))
elif isinstance(elt, str):
obj = libgap.eval(elt)
if obj is not None:
if not isinstance(obj, (GapElement_Integer, GapElement_Rational,
GapElement_Cyclotomic)):
raise TypeError(
"{} of type {} not valid to initialize an element of the universal cyclotomic field"
.format(obj, type(obj)))
return self.element_class(self, obj)
# late import to avoid slowing down the above conversions
from sage.rings.number_field.number_field_element import NumberFieldElement
from sage.rings.number_field.number_field import NumberField_cyclotomic, CyclotomicField
P = parent(elt)
if isinstance(elt, NumberFieldElement) and isinstance(
P, NumberField_cyclotomic):
n = P.gen().multiplicative_order()
elt = CyclotomicField(n)(elt)
return sum(c * self.gen(n, i)
for i, c in enumerate(elt._coefficients()))
if hasattr(elt, '_algebraic_'):
return elt._algebraic_(self)
raise TypeError(
"{} of type {} not valid to initialize an element of the universal cyclotomic field"
.format(elt, type(elt)))
def molien_series(self,
chi=None,
return_series=True,
prec=20,
variable='t'):
r"""
Compute the Molien series of this finite group with respect to the
character ``chi``. It can be returned either as a rational function
in one variable or a power series in one variable. The base field
must be a finite field, the rationals, or a cyclotomic field.
Note that the base field characteristic cannot divide the group
order (i.e., the non-modular case).
ALGORITHM:
For a finite group `G` in characteristic zero we construct the Molien series as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\text{det}(I-tg)},
where `I` is the identity matrix and `t` an indeterminate.
For characteristic `p` not dividing the order of `G`, let `k` be the base field
and `N` the order of `G`. Define `\lambda` as a primitive `N`-th root of unity over `k`
and `\omega` as a primitive `N`-th root of unity over `\QQ`. For each `g \in G`
define `k_i(g)` to be the positive integer such that
`e_i = \lambda^{k_i(g)}` for each eigenvalue `e_i` of `g`. Then the Molien series
is computed as
.. MATH::
\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\prod_{i=1}^n(1 - t\omega^{k_i(g)})},
where `t` is an indeterminant. [Dec1998]_
INPUT:
- ``chi`` -- (default: trivial character) a linear group character of this group
- ``return_series`` -- boolean (default: ``True``) if ``True``, then returns
the Molien series as a power series, ``False`` as a rational function
- ``prec`` -- integer (default: 20); power series default precision
- ``variable`` -- string (default: ``'t'``); Variable name for the Molien series
OUTPUT: single variable rational function or power series with integer coefficients
EXAMPLES::
sage: MatrixGroup(matrix(QQ,2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups
sage: MatrixGroup(matrix(GF(3),2,2,[1,1,0,1])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: characteristic cannot divide group order
Tetrahedral Group::
sage: K.<i> = CyclotomicField(4)
sage: Tetra = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [0,i, -i,0])
sage: Tetra.molien_series(prec=30)
1 + t^8 + 2*t^12 + t^16 + 2*t^20 + 3*t^24 + 2*t^28 + O(t^30)
sage: mol = Tetra.molien_series(return_series=False); mol
(t^8 - t^4 + 1)/(t^16 - t^12 - t^4 + 1)
sage: mol.parent()
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
sage: chi = Tetra.character(Tetra.character_table()[1])
sage: Tetra.molien_series(chi, prec=30, variable='u')
u^6 + u^14 + 2*u^18 + u^22 + 2*u^26 + 3*u^30 + 2*u^34 + O(u^36)
sage: chi = Tetra.character(Tetra.character_table()[2])
sage: Tetra.molien_series(chi)
t^10 + t^14 + t^18 + 2*t^22 + 2*t^26 + O(t^30)
::
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: mol = S3.molien_series(prec=10); mol
1 + t + 2*t^2 + 3*t^3 + 4*t^4 + 5*t^5 + 7*t^6 + 8*t^7 + 10*t^8 + 12*t^9 + O(t^10)
sage: mol.parent()
Power Series Ring in t over Integer Ring
Octahedral Group::
sage: K.<v> = CyclotomicField(8)
sage: a = v-v^3 #sqrt(2)
sage: i = v^2
sage: Octa = MatrixGroup([(-1+i)/2,(-1+i)/2, (1+i)/2,(-1-i)/2], [(1+i)/a,0, 0,(1-i)/a])
sage: Octa.molien_series(prec=30)
1 + t^8 + t^12 + t^16 + t^18 + t^20 + 2*t^24 + t^26 + t^28 + O(t^30)
Icosahedral Group::
sage: K.<v> = CyclotomicField(10)
sage: z5 = v^2
sage: i = z5^5
sage: a = 2*z5^3 + 2*z5^2 + 1 #sqrt(5)
sage: Ico = MatrixGroup([[z5^3,0, 0,z5^2], [0,1, -1,0], [(z5^4-z5)/a, (z5^2-z5^3)/a, (z5^2-z5^3)/a, -(z5^4-z5)/a]])
sage: Ico.molien_series(prec=40)
1 + t^12 + t^20 + t^24 + t^30 + t^32 + t^36 + O(t^40)
::
sage: G = MatrixGroup(CyclicPermutationGroup(3))
sage: chi = G.character(G.character_table()[1])
sage: G.molien_series(chi, prec=10)
t + 2*t^2 + 3*t^3 + 5*t^4 + 7*t^5 + 9*t^6 + 12*t^7 + 15*t^8 + 18*t^9 + 22*t^10 + O(t^11)
::
sage: K = GF(5)
sage: S = MatrixGroup(SymmetricGroup(4))
sage: G = MatrixGroup([matrix(K,4,4,[K(y) for u in m.list() for y in u])for m in S.gens()])
sage: G.molien_series(return_series=False)
1/(t^10 - t^9 - t^8 + 2*t^5 - t^2 - t + 1)
::
sage: i = GF(7)(3)
sage: G = MatrixGroup([[i^3,0,0,-i^3],[i^2,0,0,-i^2]])
sage: chi = G.character(G.character_table()[4])
sage: G.molien_series(chi)
3*t^5 + 6*t^11 + 9*t^17 + 12*t^23 + O(t^25)
"""
if not self.is_finite():
raise NotImplementedError("only implemented for finite groups")
if chi is None:
chi = self.trivial_character()
M = self.matrix_space()
R = FractionField(self.base_ring())
N = self.order()
if R.characteristic() == 0:
P = PolynomialRing(R, variable)
t = P.gen()
#it is possible the character is over a larger cyclotomic field
K = chi.values()[0].parent()
if K.degree() != 1:
if R.degree() != 1:
L = K.composite_fields(R)[0]
else:
L = K
else:
L = R
mol = P(0)
for g in self:
mol += L(chi(g)) / (M.identity_matrix() -
t * g.matrix()).det().change_ring(L)
elif R.characteristic().divides(N):
raise NotImplementedError(
"characteristic cannot divide group order")
else: #char p>0
#find primitive Nth roots of unity over base ring and QQ
F = cyclotomic_polynomial(N).change_ring(R)
w = F.roots(ring=R.algebraic_closure(), multiplicities=False)[0]
#don't need to extend further in this case since the order of
#the roots of unity in the character divide the order of the group
L = CyclotomicField(N, 'v')
v = L.gen()
#construct Molien series
P = PolynomialRing(L, variable)
t = P.gen()
mol = P(0)
for g in self:
#construct Phi
phi = L(chi(g))
for e in g.matrix().eigenvalues():
#find power such that w**n = e
n = 1
while w**n != e and n < N + 1:
n += 1
#raise v to that power
phi *= (1 - t * v**n)
mol += P(1) / phi
#We know the coefficients will be integers
mol = mol.numerator().change_ring(ZZ) / mol.denominator().change_ring(
ZZ)
#divide by group order
mol /= N
if return_series:
PS = PowerSeriesRing(ZZ, variable, default_prec=prec)
return PS(mol)
return mol
