def reynolds_operator(self, poly, chi=None):
r"""
Compute the Reynolds operator of this finite group `G`.
This is the projection from a polynomial ring to the ring of
relative invariants [Stu1993]_. If possible, the invariant is
returned defined over the base field of the given polynomial
``poly``, otherwise, it is returned over the compositum of the
fields involved in the computation.
Only implemented for absolute fields.
ALGORITHM:
Let `K[x]` be a polynomial ring and `\chi` a linear character for `G`. Let
.. MATH:
K[x]^G_{\chi} = \{f \in K[x] | \pi f = \chi(\pi) f \forall \pi\in G\}
be the ring of invariants of `G` relative to `\chi`. Then the Reynold's operator
is a map `R` from `K[x]` into `K[x]^G_{\chi}` defined by
.. MATH:
f \mapsto \frac{1}{|G|} \sum_{ \pi \in G} \chi(\pi) f.
INPUT:
- ``poly`` -- a polynomial
- ``chi`` -- (default: trivial character) a linear group character of this group
OUTPUT: an invariant polynomial relative to `\chi`
AUTHORS:
Rebecca Lauren Miller and Ben Hutz
EXAMPLES::
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: R.<x,y,z> = QQ[]
sage: f = x*y*z^3
sage: S3.reynolds_operator(f)
1/3*x^3*y*z + 1/3*x*y^3*z + 1/3*x*y*z^3
::
sage: G = MatrixGroup(CyclicPermutationGroup(4))
sage: chi = G.character(G.character_table()[3])
sage: K.<v> = CyclotomicField(4)
sage: R.<x,y,z,w> = K[]
sage: G.reynolds_operator(x, chi)
1/4*x + (1/4*v)*y - 1/4*z + (-1/4*v)*w
sage: chi = G.character(G.character_table()[2])
sage: R.<x,y,z,w> = QQ[]
sage: G.reynolds_operator(x*y, chi)
1/4*x*y + (-1/4*zeta4)*y*z + (1/4*zeta4)*x*w - 1/4*z*w
::
sage: K.<i> = CyclotomicField(4)
sage: G = MatrixGroup(CyclicPermutationGroup(3))
sage: chi = G.character(G.character_table()[1])
sage: R.<x,y,z> = K[]
sage: G.reynolds_operator(x*y^5, chi)
1/3*x*y^5 + (2/3*izeta3^3 + izeta3^2 + 8/3*izeta3 + 1)*x^5*z +
(-2/3*izeta3^3 - izeta3^2 - 8/3*izeta3 - 4/3)*y*z^5
sage: R.<x,y,z> = QQbar[]
sage: G.reynolds_operator(x*y^5, chi)
1/3*x*y^5 + (-0.1666666666666667? - 0.2886751345948129?*I)*x^5*z +
(-0.1666666666666667? + 0.2886751345948129?*I)*y*z^5
::
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: chi = Tetra.character(Tetra.character_table()[4])
sage: L.<v> = QuadraticField(-3)
sage: R.<x,y> = L[]
sage: Tetra.reynolds_operator(x^4)
0
sage: Tetra.reynolds_operator(x^4, chi)
1/4*x^4 + (1/2*v)*x^2*y^2 + 1/4*y^4
sage: R.<x>=L[]
sage: LL.<w> = L.extension(x^2+v)
sage: R.<x,y> = LL[]
sage: Tetra.reynolds_operator(x^4, chi)
Traceback (most recent call last):
...
NotImplementedError: only implemented for absolute fields
::
sage: G = MatrixGroup(DihedralGroup(4))
sage: chi = G.character(G.character_table()[1])
sage: R.<x,y> = QQ[]
sage: f = x^4
sage: G.reynolds_operator(f, chi)
Traceback (most recent call last):
...
TypeError: number of variables in polynomial must match size of matrices
sage: R.<x,y,z,w> = QQ[]
sage: f = x^3*y
sage: G.reynolds_operator(f, chi)
1/8*x^3*y - 1/8*x*y^3 + 1/8*y^3*z - 1/8*y*z^3 - 1/8*x^3*w + 1/8*z^3*w +
1/8*x*w^3 - 1/8*z*w^3
Characteristic p>0 examples::
sage: G = MatrixGroup([[0,1,1,0]])
sage: R.<w,x> = GF(2)[]
sage: G.reynolds_operator(x)
Traceback (most recent call last):
...
NotImplementedError: not implemented when characteristic divides group order
::
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: R.<w,x> = GF(7)[]
sage: f = w^5*x + x^6
sage: G.reynolds_operator(f, chi)
Traceback (most recent call last):
...
NotImplementedError: nontrivial characters not implemented for characteristic > 0
sage: G.reynolds_operator(f)
x^6
::
sage: K = GF(3^2,'t')
sage: G = MatrixGroup([matrix(K,2,2, [0,K.gen(),1,0])])
sage: R.<x,y> = GF(3)[]
sage: G.reynolds_operator(x^8)
-x^8 - y^8
::
sage: K = GF(3^2,'t')
sage: G = MatrixGroup([matrix(GF(3),2,2, [0,1,1,0])])
sage: R.<x,y> = K[]
sage: f = -K.gen()*x
sage: G.reynolds_operator(f)
(t)*x + (t)*y
"""
if poly.parent().ngens() != self.degree():
raise TypeError("number of variables in polynomial must match size of matrices")
R = FractionField(poly.base_ring())
C = FractionField(self.base_ring())
if chi is None: #then this is the trivial character
if R.characteristic() == 0:
#non-modular case
if C == QQbar or R == QQbar:
L = QQbar
elif not C.is_absolute() or not R.is_absolute():
raise NotImplementedError("only implemented for absolute fields")
else: #create the compositum
if C.absolute_degree() == 1:
L = R
elif R.absolute_degree() == 1:
L = C
else:
L = C.composite_fields(R)[0]
elif not R.characteristic().divides(self.order()):
if R.characteristic() != C.characteristic():
raise ValueError("base fields must have same characteristic")
else:
if R.degree() >= C.degree():
L = R
else:
L = C
else:
raise NotImplementedError("not implemented when characteristic divides group order")
poly = poly.change_ring(L)
poly_gens = vector(poly.parent().gens())
F = L.zero()
for g in self:
F += poly(*g.matrix()*vector(poly.parent().gens()))
F /= self.order()
return F
#non-trivial character case
K = chi.values()[0].parent()
if R.characteristic() == 0:
#extend base_ring to compositum
if C == QQbar or K == QQbar or R == QQbar:
L = QQbar
elif not C.is_absolute() or not K.is_absolute() or not R.is_absolute():
raise NotImplementedError("only implemented for absolute fields")
else:
fields = []
for M in [R,K,C]:
if M.absolute_degree() != 1:
fields.append(M)
l = len(fields)
if l == 0:
# all are QQ
L = R
elif l == 1:
#only one is an extension
L = fields[0]
elif l == 2:
#only two are extensions
L = fields[0].composite_fields(fields[1])[0]
else:
#all three are extensions
L1 = fields[0].composite_fields(fields[1])[0]
L = L1.composite_fields(fields[2])[0]
else:
raise NotImplementedError("nontrivial characters not implemented for characteristic > 0")
poly = poly.change_ring(L)
poly_gens = vector(poly.parent().gens())
F = L.zero()
for g in self:
F += L(chi(g)) * poly(*g.matrix().change_ring(L)*poly_gens)
F /= self.order()
try: # attempt to move F to base_ring of polyomial
F = F.change_ring(R)
except (TypeError, ValueError):
pass
return F
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 reynolds_operator(self, poly, chi=None):
r"""
Compute the Reynolds operator of this finite group `G`.
This is the projection from a polynomial ring to the ring of
relative invariants [Stu1993]_. If possible, the invariant is
returned defined over the base field of the given polynomial
``poly``, otherwise, it is returned over the compositum of the
fields involved in the computation.
Only implemented for absolute fields.
ALGORITHM:
Let `K[x]` be a polynomial ring and `\chi` a linear character for `G`. Let
.. MATH:
K[x]^G_{\chi} = \{f \in K[x] | \pi f = \chi(\pi) f \forall \pi\in G\}
be the ring of invariants of `G` relative to `\chi`. Then the Reynold's operator
is a map `R` from `K[x]` into `K[x]^G_{\chi}` defined by
.. MATH:
f \mapsto \frac{1}{|G|} \sum_{ \pi \in G} \chi(\pi) f.
INPUT:
- ``poly`` -- a polynomial
- ``chi`` -- (default: trivial character) a linear group character of this group
OUTPUT: an invariant polynomial relative to `\chi`
AUTHORS:
Rebecca Lauren Miller and Ben Hutz
EXAMPLES::
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: R.<x,y,z> = QQ[]
sage: f = x*y*z^3
sage: S3.reynolds_operator(f)
1/3*x^3*y*z + 1/3*x*y^3*z + 1/3*x*y*z^3
::
sage: G = MatrixGroup(CyclicPermutationGroup(4))
sage: chi = G.character(G.character_table()[3])
sage: K.<v> = CyclotomicField(4)
sage: R.<x,y,z,w> = K[]
sage: G.reynolds_operator(x, chi)
1/4*x + (-1/4*v)*y - 1/4*z + (1/4*v)*w
sage: chi = G.character(G.character_table()[2])
sage: R.<x,y,z,w> = QQ[]
sage: G.reynolds_operator(x*y, chi)
1/4*x*y + (1/4*zeta4)*y*z + (-1/4*zeta4)*x*w - 1/4*z*w
::
sage: K.<i> = CyclotomicField(4)
sage: G = MatrixGroup(CyclicPermutationGroup(3))
sage: chi = G.character(G.character_table()[1])
sage: R.<x,y,z> = K[]
sage: G.reynolds_operator(x*y^5, chi)
1/3*x*y^5 + (2/3*izeta3^3 + izeta3^2 + 8/3*izeta3 + 1)*x^5*z + (-2/3*izeta3^3 - izeta3^2 - 8/3*izeta3 - 4/3)*y*z^5
sage: R.<x,y,z> = QQbar[]
sage: G.reynolds_operator(x*y^5, chi)
1/3*x*y^5 + (-0.1666666666666667? - 0.2886751345948129?*I)*x^5*z + (-0.1666666666666667? + 0.2886751345948129?*I)*y*z^5
::
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: chi = Tetra.character(Tetra.character_table()[4])
sage: L.<v> = QuadraticField(-3)
sage: R.<x,y> = L[]
sage: Tetra.reynolds_operator(x^4)
0
sage: Tetra.reynolds_operator(x^4, chi)
1/4*x^4 + (1/2*v)*x^2*y^2 + 1/4*y^4
sage: R.<x>=L[]
sage: LL.<w> = L.extension(x^2+v)
sage: R.<x,y> = LL[]
sage: Tetra.reynolds_operator(x^4, chi)
Traceback (most recent call last):
...
NotImplementedError: only implemented for absolute fields
::
sage: G = MatrixGroup(DihedralGroup(4))
sage: chi = G.character(G.character_table()[1])
sage: R.<x,y> = QQ[]
sage: f = x^4
sage: G.reynolds_operator(f, chi)
Traceback (most recent call last):
...
TypeError: number of variables in polynomial must match size of matrices
sage: R.<x,y,z,w> = QQ[]
sage: f = x^3*y
sage: G.reynolds_operator(f, chi)
1/8*x^3*y - 1/8*x*y^3 + 1/8*y^3*z - 1/8*y*z^3 - 1/8*x^3*w + 1/8*z^3*w +
1/8*x*w^3 - 1/8*z*w^3
Characteristic p>0 examples::
sage: G = MatrixGroup([[0,1,1,0]])
sage: R.<w,x> = GF(2)[]
sage: G.reynolds_operator(x)
Traceback (most recent call last):
...
NotImplementedError: not implemented when characteristic divides group order
::
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: R.<w,x> = GF(7)[]
sage: f = w^5*x + x^6
sage: G.reynolds_operator(f, chi)
Traceback (most recent call last):
...
NotImplementedError: nontrivial characters not implemented for characteristic > 0
sage: G.reynolds_operator(f)
x^6
::
sage: K = GF(3^2,'t')
sage: G = MatrixGroup([matrix(K,2,2, [0,K.gen(),1,0])])
sage: R.<x,y> = GF(3)[]
sage: G.reynolds_operator(x^8)
-x^8 - y^8
::
sage: K = GF(3^2,'t')
sage: G = MatrixGroup([matrix(GF(3),2,2, [0,1,1,0])])
sage: R.<x,y> = K[]
sage: f = -K.gen()*x
sage: G.reynolds_operator(f)
(t)*x + (t)*y
"""
if poly.parent().ngens() != self.degree():
raise TypeError(
"number of variables in polynomial must match size of matrices"
)
R = FractionField(poly.base_ring())
C = FractionField(self.base_ring())
if chi is None: #then this is the trivial character
if R.characteristic() == 0:
#non-modular case
if C == QQbar or R == QQbar:
L = QQbar
elif not C.is_absolute() or not R.is_absolute():
raise NotImplementedError(
"only implemented for absolute fields")
else: #create the compositum
if C.absolute_degree() == 1:
L = R
elif R.absolute_degree() == 1:
L = C
else:
L = C.composite_fields(R)[0]
elif not R.characteristic().divides(self.order()):
if R.characteristic() != C.characteristic():
raise ValueError(
"base fields must have same characteristic")
else:
if R.degree() >= C.degree():
L = R
else:
L = C
else:
raise NotImplementedError(
"not implemented when characteristic divides group order")
poly = poly.change_ring(L)
poly_gens = vector(poly.parent().gens())
F = L.zero()
for g in self:
F += poly(*g.matrix() * vector(poly.parent().gens()))
F /= self.order()
return F
#non-trivial character case
K = chi.values()[0].parent()
if R.characteristic() == 0:
#extend base_ring to compositum
if C == QQbar or K == QQbar or R == QQbar:
L = QQbar
elif not C.is_absolute() or not K.is_absolute(
) or not R.is_absolute():
raise NotImplementedError(
"only implemented for absolute fields")
else:
fields = []
for M in [R, K, C]:
if M.absolute_degree() != 1:
fields.append(M)
l = len(fields)
if l == 0:
# all are QQ
L = R
elif l == 1:
#only one is an extension
L = fields[0]
elif l == 2:
#only two are extensions
L = fields[0].composite_fields(fields[1])[0]
else:
#all three are extensions
L1 = fields[0].composite_fields(fields[1])[0]
L = L1.composite_fields(fields[2])[0]
else:
raise NotImplementedError(
"nontrivial characters not implemented for characteristic > 0")
poly = poly.change_ring(L)
poly_gens = vector(poly.parent().gens())
F = L.zero()
for g in self:
F += L(chi(g)) * poly(*g.matrix().change_ring(L) * poly_gens)
F /= self.order()
try: # attempt to move F to base_ring of polynomial
F = F.change_ring(R)
except (TypeError, ValueError):
pass
return F
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 binary_quintic_coefficients_from_invariants(invariants, K=None, invariant_choice='default', scaling='none'):
r"""
Reconstruct a binary quintic from the values of its (Clebsch) invariants.
INPUT:
- ``invariants`` -- A list or tuple of values of the three or four
invariants. The default option requires the Clebsch invariants `A`, `B`,
`C` and `R` of the binary quintic.
- ``K`` -- The field over which the quintic is defined.
- ``invariant_choice`` -- The type of invariants provided. The accepted
options are ``'clebsch'`` and ``'default'``, which are the same. No
other options are implemented.
- ``scaling`` -- How the coefficients should be scaled. The accepted
values are ``'none'`` for no scaling, ``'normalized'`` to scale in such
a way that the resulting coefficients are independent of the scaling of
the input invariants and ``'coprime'`` which scales the input invariants
by dividing them by their gcd.
OUTPUT:
A set of coefficients of a binary quintic, whose invariants are equal to
the given ``invariants`` up to a scaling.
EXAMPLES:
First we check the general case, where the invariant `M` is non-zero::
sage: R.<x0, x1> = QQ[]
sage: p = 3*x1^5 + 6*x1^4*x0 + 3*x1^3*x0^2 + 4*x1^2*x0^3 - 5*x1*x0^4 + 4*x0^5
sage: quintic = invariant_theory.binary_quintic(p, x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: reconstructed = invariant_theory.binary_form_from_invariants(5, invs, variables=quintic.variables()) # indirect doctest
sage: reconstructed
Binary quintic with coefficients (9592267437341790539005557/244140625000000,
2149296928207625556323004064707/610351562500000000,
11149651890347700974453304786783/76293945312500000,
122650775751894638395648891202734239/47683715820312500000,
323996630945706528474286334593218447/11920928955078125000,
1504506503644608395841632538558481466127/14901161193847656250000)
We can see that the invariants of the reconstructed form match the ones of
the original form by scaling the invariants `B` and `C`::
sage: scale = invs[0]/reconstructed.A_invariant()
sage: invs[1] == reconstructed.B_invariant()*scale^2
True
sage: invs[2] == reconstructed.C_invariant()*scale^3
True
If we compare the form obtained by this reconstruction to the one found by
letting the covariants `\alpha` and `\beta` be the coordinates of the form,
we find the forms are the same up to a power of the determinant of `\alpha`
and `\beta`::
sage: alpha = quintic.alpha_covariant()
sage: beta = quintic.beta_covariant()
sage: g = matrix([[alpha(x0=1,x1=0),alpha(x0=0,x1=1)],[beta(x0=1,x1=0),beta(x0=0,x1=1)]])^-1
sage: transformed = tuple([g.determinant()^-5*x for x in quintic.transformed(g).coeffs()])
sage: transformed == reconstructed.coeffs()
True
This can also be seen by computing the `\alpha` covariant of the obtained
form::
sage: reconstructed.alpha_covariant().coefficient(x1)
0
sage: reconstructed.alpha_covariant().coefficient(x0) != 0
True
If the invariant `M` vanishes, then the coefficients are computed in a
different way::
sage: [A,B,C] = [3,1,2]
sage: M = 2*A*B - 3*C
sage: M
0
sage: from sage.rings.invariants.reconstruction import binary_quintic_coefficients_from_invariants
sage: reconstructed = binary_quintic_coefficients_from_invariants([A,B,C])
sage: reconstructed
(-66741943359375/2097152,
-125141143798828125/134217728,
0,
52793920040130615234375/34359738368,
19797720015048980712890625/1099511627776,
-4454487003386020660400390625/17592186044416)
sage: newform = sum([ reconstructed[i]*x0^i*x1^(5-i) for i in range(6) ])
sage: newquintic = invariant_theory.binary_quintic(newform, x0, x1)
sage: scale = 3/newquintic.A_invariant()
sage: [3, newquintic.B_invariant()*scale^2, newquintic.C_invariant()*scale^3]
[3, 1, 2]
Several special cases::
sage: quintic = invariant_theory.binary_quintic(x0^5 - x1^5, x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 0, 0, 1)
sage: quintic = invariant_theory.binary_quintic(x0*x1*(x0^3-x1^3), x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(0, 1, 0, 0, 1, 0)
sage: quintic = invariant_theory.binary_quintic(x0^5 + 10*x0^3*x1^2 - 15*x0*x1^4, x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 10, 0, -15, 0)
sage: quintic = invariant_theory.binary_quintic(x0^2*(x0^3 + x1^3), x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 1, 0, 0)
sage: quintic = invariant_theory.binary_quintic(x0*(x0^4 + x1^4), x0, x1)
sage: invs = quintic.clebsch_invariants(as_tuple=True)
sage: binary_quintic_coefficients_from_invariants(invs)
(1, 0, 0, 0, 1, 0)
For fields of characteristic 2, 3 or 5, there is no reconstruction
implemented. This is part of :trac:`26786`.::
sage: binary_quintic_coefficients_from_invariants([3,1,2], K=GF(5))
Traceback (most recent call last):
...
NotImplementedError: no reconstruction of binary quintics implemented for fields of characteristic 2, 3 or 5
TESTS::
sage: from sage.rings.invariants.reconstruction import binary_quintic_coefficients_from_invariants
sage: binary_quintic_coefficients_from_invariants([1,2,3], scaling='unknown')
Traceback (most recent call last):
...
ValueError: unknown scaling option 'unknown'
"""
if invariant_choice not in ['default', 'clebsch']:
raise ValueError('unknown choice of invariants {} for a binary quintic'
.format(invariant_choice))
if scaling not in ['none', 'normalized', 'coprime']:
raise ValueError("unknown scaling option '%s'" % scaling)
if scaling == 'coprime':
if len(invariants) == 3:
invariants = _reduce_invariants(invariants, [1,2,3])
elif len(invariants) == 4:
invariants = _reduce_invariants(invariants, [2,4,6,9])
A, B, C = invariants[0:3]
if K is None:
from sage.rings.fraction_field import FractionField
K = FractionField(A.parent())
if K.characteristic() in [2, 3, 5]:
raise NotImplementedError('no reconstruction of binary quintics '
'implemented for fields of characteristic 2, 3 or 5')
M = 2*A*B - 3*C
N = K(2)**-1 * (A*C-B**2)
R2 = -K(2)**-1 * (A*N**2-2*B*M*N+C*M**2)
scale = [1,1,1,1,1,1]
from sage.functions.all import binomial, sqrt
if len(invariants) == 3:
if R2.is_square():
R = sqrt(R2)
else:
# if R2 is not a square, we scale the invariants in a suitable way
# so that the 'new' R2 is a square
[A, B, C] = [R2*A, R2**2*B, R2**3*C]
[M, N] = [R2**3*M, R2**4*N]
R = R2**5
elif len(invariants) == 4:
if invariants[3]**2 != R2:
raise ValueError('provided invariants do not satisfy the syzygy '
'for Clebsch invariants of a binary quintic')
R = invariants[3]
else:
raise ValueError('incorrect number of invariants provided, this '
'method requires 3 or 4 invariants')
if M == 0:
if N == 0:
if A == 0:
raise ValueError('no unique reconstruction possible for '
'quintics with a treefold linear factor')
else:
if B == 0:
return (1,0,0,0,0,1)
else:
return (0,1,0,0,1,0)
else:
# case corresponding to using alpha and gamma as coordinates
if A == 0:
return (1,0,0,0,1,0)
else:
if scaling == 'normalized':
# scaling z by (R/A**3)
scale = [ (-N)**-5*A**6*(R/A**3)**i for i in range(6) ]
D = -N
Delta = C
a = [0]
a.append((2*K(3)**-1*A**2-B)*N*B*K(2)**-1 - N**2*K(2)**-1)
B0 = 2*K(3)**-1*A*R
B1 = A*N*B*K(3)**-1
C0 = 2*K(3)**-1*R
C1 = B*N
else:
# case corresponding to using alpha and beta as coordinates
if R == 0:
if A == 0:
return (1,0,10,0,-15,0)
elif scaling == 'normalized':
# scaling x by A and z by sqrt(A)
scale = [ (-M)**(-5)*sqrt(A)**(12+i) for i in range(6) ]
else:
if A == 0:
if B == 0:
return (1,0,0,1,0,0)
elif scaling == 'normalized':
# scaling y by R/B**2
scale = [ (-M)**(-3)*(R/B**2)**i for i in range(6) ]
elif scaling == 'normalized':
# scaling y by R/A**4
scale = [ (-M)**(-3)*(R/A**4)**i for i in range(6) ]
D = -M
Delta = A
a = [0]
a.append((2*K(3)**-1*A**2-B)*(N*A-M*B)*K(2)**-1 \
- M*(N*K(2)**-1-M*A*K(3)**-1))
B0 = R
B1 = K(2)**-1*(N*A-M*B)
C0 = 0
C1 = -M
a[0] = (2*K(3)**-1*A**2-B)*R
a.append(-D*B0 - K(2)**-1*Delta*a[0])
a.append(-D*B1 - K(2)**-1*Delta*a[1])
a.append(D**2*C0 + D*Delta*B0 + K(4)**-1*Delta**2*a[0])
a.append(D**2*C1 + D*Delta*B1 + K(4)**-1*Delta**2*a[1])
coeffs = tuple([K((-1)**i*binomial(5,i)*scale[5-i]*a[i]) for i in range(6)])
if scaling == 'coprime':
from sage.arith.misc import gcd
return tuple([coeffs[i]/gcd(coeffs) for i in range(6)])
else:
return coeffs
