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 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
