Finitely Generated Matrix Groups with GAP#

class sage.groups.matrix_gps.finitely_generated_gap.FinitelyGeneratedMatrixGroup_gap(degree, base_ring, libgap_group, ambient=None, category=None)[source]#

Bases: MatrixGroup_gap

Matrix group generated by a finite number of matrices.

EXAMPLES:

sage: m1 = matrix(GF(11), [[1,2],[3,4]])
sage: m2 = matrix(GF(11), [[1,3],[10,0]])
sage: G = MatrixGroup(m1, m2);  G
Matrix group over Finite Field of size 11 with 2 generators (
[1 2]  [ 1  3]
[3 4], [10  0]
)
sage: type(G)
<class 'sage.groups.matrix_gps.finitely_generated_gap.FinitelyGeneratedMatrixGroup_gap_with_category'>
sage: TestSuite(G).run()

>>> from sage.all import *
>>> m1 = matrix(GF(Integer(11)), [[Integer(1),Integer(2)],[Integer(3),Integer(4)]])
>>> m2 = matrix(GF(Integer(11)), [[Integer(1),Integer(3)],[Integer(10),Integer(0)]])
>>> G = MatrixGroup(m1, m2);  G
Matrix group over Finite Field of size 11 with 2 generators (
[1 2]  [ 1  3]
[3 4], [10  0]
)
>>> type(G)
<class 'sage.groups.matrix_gps.finitely_generated_gap.FinitelyGeneratedMatrixGroup_gap_with_category'>
>>> TestSuite(G).run()
as_permutation_group(algorithm=None, seed=None)[source]#

Return a permutation group representation for the group.

In most cases occurring in practice, this is a permutation group of minimal degree (the degree being determined from orbits under the group action). When these orbits are hard to compute, the procedure can be time-consuming and the degree may not be minimal.

INPUT:

  • algorithm – None or 'smaller'; in the latter case, try harder to find a permutation representation of small degree

  • seed – None or an integer specifying the seed to fix results depending on pseudo-random-numbers; here it makes sense to be used with respect to the 'smaller' option, since GAP produces random output in that context

OUTPUT:

A permutation group isomorphic to self. The algorithm='smaller' option tries to return an isomorphic group of low degree, but is not guaranteed to find the smallest one and must not even differ from the one obtained without the option. In that case repeating the invocation may help (see the example below).

EXAMPLES:

sage: MS = MatrixSpace(GF(2), 5, 5)
sage: A = MS([[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]])
sage: G = MatrixGroup([A])
sage: G.as_permutation_group().order()
2

>>> from sage.all import *
>>> MS = MatrixSpace(GF(Integer(2)), Integer(5), Integer(5))
>>> A = MS([[Integer(0),Integer(0),Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(0),Integer(0),Integer(1),Integer(0)],[Integer(0),Integer(0),Integer(1),Integer(0),Integer(0)],[Integer(0),Integer(1),Integer(0),Integer(0),Integer(0)],[Integer(1),Integer(0),Integer(0),Integer(0),Integer(0)]])
>>> G = MatrixGroup([A])
>>> G.as_permutation_group().order()
2

A finite subgroup of \(GL(12,\ZZ)\) as a permutation group:

sage: imf = libgap.function_factory('ImfMatrixGroup')
sage: GG = imf( 12, 3 )
sage: G = MatrixGroup(GG.GeneratorsOfGroup())
sage: G.cardinality()
21499084800
sage: P = G.as_permutation_group()
sage: Psmaller = G.as_permutation_group(algorithm="smaller", seed=6)
sage: P.cardinality()
21499084800
sage: P.degree()
144
sage: Psmaller.cardinality()
21499084800
sage: Psmaller.degree() <= P.degree()
True

>>> from sage.all import *
>>> imf = libgap.function_factory('ImfMatrixGroup')
>>> GG = imf( Integer(12), Integer(3) )
>>> G = MatrixGroup(GG.GeneratorsOfGroup())
>>> G.cardinality()
21499084800
>>> P = G.as_permutation_group()
>>> Psmaller = G.as_permutation_group(algorithm="smaller", seed=Integer(6))
>>> P.cardinality()
21499084800
>>> P.degree()
144
>>> Psmaller.cardinality()
21499084800
>>> Psmaller.degree() <= P.degree()
True

Note

In this case, the “smaller” option returned an isomorphic group of lower degree. The above example used GAP’s library of irreducible maximal finite (“imf”) integer matrix groups to construct the MatrixGroup \(G\) over \(\GF{7}\). The section “Irreducible Maximal Finite Integral Matrix Groups” in the GAP reference manual has more details.

Note

Concerning the option algorithm='smaller' you should note the following from GAP documentation: “The methods used might involve the use of random elements and the permutation representation (or even the degree of the representation) is not guaranteed to be the same for different calls of SmallerDegreePermutationRepresentation.”

To obtain a reproducible result the optional argument seed may be used as in the example above.

invariant_generators()[source]#

Return invariant ring generators.

Computes generators for the polynomial ring \(F[x_1,\ldots,x_n]^G\), where \(G\) in \(GL(n,F)\) is a finite matrix group.

In the “good characteristic” case the polynomials returned form a minimal generating set for the algebra of \(G\)-invariant polynomials. In the “bad” case, the polynomials returned are primary and secondary invariants, forming a not necessarily minimal generating set for the algebra of \(G\)-invariant polynomials.

ALGORITHM:

Wraps Singular’s invariant_algebra_reynolds and invariant_ring in finvar.lib.

EXAMPLES:

sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]),MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.invariant_generators()                                              # needs sage.libs.singular
[x1^7*x2 - x1*x2^7,
 x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12,
 x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]

sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a, (q-1)/a], [1/a, -1/a]]
sage: gen2 = [[1,0], [0,-1]]; gen3 = [[-1,0], [0,1]]
sage: G = MatrixGroup([MS(gen1), MS(gen2), MS(gen3)])
sage: G.cardinality()
12
sage: G.invariant_generators()                                              # needs sage.libs.singular
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]

sage: # needs sage.rings.number_field
sage: F = CyclotomicField(8)
sage: z = F.gen()
sage: a = z+1/z
sage: b = z^2
sage: MS = MatrixSpace(F,2,2)
sage: g1 = MS([[1/a, 1/a], [1/a, -1/a]])
sage: g2 = MS([[-b, 0], [0, b]])
sage: G = MatrixGroup([g1,g2])
sage: G.invariant_generators()                                              # needs sage.libs.singular
[x1^4 + 2*x1^2*x2^2 + x2^4,
 x1^5*x2 - x1*x2^5,
 x1^8 + 28/9*x1^6*x2^2 + 70/9*x1^4*x2^4 + 28/9*x1^2*x2^6 + x2^8]

>>> from sage.all import *
>>> F = GF(Integer(7)); MS = MatrixSpace(F,Integer(2),Integer(2))
>>> gens = [MS([[Integer(0),Integer(1)],[-Integer(1),Integer(0)]]),MS([[Integer(1),Integer(1)],[Integer(2),Integer(3)]])]
>>> G = MatrixGroup(gens)
>>> G.invariant_generators()                                              # needs sage.libs.singular
[x1^7*x2 - x1*x2^7,
 x1^12 - 2*x1^9*x2^3 - x1^6*x2^6 + 2*x1^3*x2^9 + x2^12,
 x1^18 + 2*x1^15*x2^3 + 3*x1^12*x2^6 + 3*x1^6*x2^12 - 2*x1^3*x2^15 + x2^18]

>>> q = Integer(4); a = Integer(2)
>>> MS = MatrixSpace(QQ, Integer(2), Integer(2))
>>> gen1 = [[Integer(1)/a, (q-Integer(1))/a], [Integer(1)/a, -Integer(1)/a]]
>>> gen2 = [[Integer(1),Integer(0)], [Integer(0),-Integer(1)]]; gen3 = [[-Integer(1),Integer(0)], [Integer(0),Integer(1)]]
>>> G = MatrixGroup([MS(gen1), MS(gen2), MS(gen3)])
>>> G.cardinality()
12
>>> G.invariant_generators()                                              # needs sage.libs.singular
[x1^2 + 3*x2^2, x1^6 + 15*x1^4*x2^2 + 15*x1^2*x2^4 + 33*x2^6]

>>> # needs sage.rings.number_field
>>> F = CyclotomicField(Integer(8))
>>> z = F.gen()
>>> a = z+Integer(1)/z
>>> b = z**Integer(2)
>>> MS = MatrixSpace(F,Integer(2),Integer(2))
>>> g1 = MS([[Integer(1)/a, Integer(1)/a], [Integer(1)/a, -Integer(1)/a]])
>>> g2 = MS([[-b, Integer(0)], [Integer(0), b]])
>>> G = MatrixGroup([g1,g2])
>>> G.invariant_generators()                                              # needs sage.libs.singular
[x1^4 + 2*x1^2*x2^2 + x2^4,
 x1^5*x2 - x1*x2^5,
 x1^8 + 28/9*x1^6*x2^2 + 70/9*x1^4*x2^4 + 28/9*x1^2*x2^6 + x2^8]

AUTHORS:

  • David Joyner, Simon King and Martin Albrecht.

REFERENCES:

  • Singular reference manual

  • [Stu1993]

  • S. King, “Minimal Generating Sets of non-modular invariant rings of finite groups”, arXiv math/0703035.

invariants_of_degree(deg, chi=None, R=None)[source]#

Return the (relative) invariants of given degree for this group.

For this group, compute the invariants of degree deg with respect to the group character chi. The method is to project each possible monomial of degree deg via the Reynolds operator. Note that if the polynomial ring R is specified it’s base ring may be extended if the resulting invariant is defined over a bigger field.

INPUT:

  • degree – a positive integer

  • chi – (default: trivial character) a linear group character of this group

  • R – (optional) a polynomial ring

OUTPUT: list of polynomials

EXAMPLES:

sage: # needs sage.groups sage.rings.number_field
sage: Gr = MatrixGroup(SymmetricGroup(2))
sage: sorted(Gr.invariants_of_degree(3))
[x0^2*x1 + x0*x1^2, x0^3 + x1^3]
sage: R.<x,y> = QQ[]
sage: sorted(Gr.invariants_of_degree(4, R=R))
[x^2*y^2, x^3*y + x*y^3, x^4 + y^4]

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> Gr = MatrixGroup(SymmetricGroup(Integer(2)))
>>> sorted(Gr.invariants_of_degree(Integer(3)))
[x0^2*x1 + x0*x1^2, x0^3 + x1^3]
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> sorted(Gr.invariants_of_degree(Integer(4), R=R))
[x^2*y^2, x^3*y + x*y^3, x^4 + y^4]

sage: # needs sage.groups sage.rings.number_field
sage: R.<x,y,z> = QQ[]
sage: Gr = MatrixGroup(DihedralGroup(3))
sage: ct = Gr.character_table()
sage: chi = Gr.character(ct[0])
sage: all(f(*(g.matrix()*vector(R.gens()))) == chi(g)*f
....: for f in Gr.invariants_of_degree(3, R=R, chi=chi) for g in Gr)
True

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> Gr = MatrixGroup(DihedralGroup(Integer(3)))
>>> ct = Gr.character_table()
>>> chi = Gr.character(ct[Integer(0)])
>>> all(f(*(g.matrix()*vector(R.gens()))) == chi(g)*f
... for f in Gr.invariants_of_degree(Integer(3), R=R, chi=chi) for g in Gr)
True

sage: i = GF(7)(3)
sage: G = MatrixGroup([[i^3,0,0,-i^3],[i^2,0,0,-i^2]])
sage: G.invariants_of_degree(25)                                            # needs sage.rings.number_field
[]

>>> from sage.all import *
>>> i = GF(Integer(7))(Integer(3))
>>> G = MatrixGroup([[i**Integer(3),Integer(0),Integer(0),-i**Integer(3)],[i**Integer(2),Integer(0),Integer(0),-i**Integer(2)]])
>>> G.invariants_of_degree(Integer(25))                                            # needs sage.rings.number_field
[]

sage: # needs sage.groups
sage: G = MatrixGroup(SymmetricGroup(5))
sage: R = QQ['x,y']
sage: G.invariants_of_degree(3, R=R)
Traceback (most recent call last):
...
TypeError: number of variables in polynomial ring must match size of matrices

>>> from sage.all import *
>>> # needs sage.groups
>>> G = MatrixGroup(SymmetricGroup(Integer(5)))
>>> R = QQ['x,y']
>>> G.invariants_of_degree(Integer(3), R=R)
Traceback (most recent call last):
...
TypeError: number of variables in polynomial ring must match size of matrices

sage: # needs sage.groups sage.rings.number_field
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: sorted(G.invariants_of_degree(2, R=R, chi=chi))
[x*y + (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*x*z
     + (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*y*z,
 x^2 + (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*y^2
     + (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*z^2]

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> K = CyclotomicField(Integer(4), names=('i',)); (i,) = K._first_ngens(1)
>>> G =  MatrixGroup(CyclicPermutationGroup(Integer(3)))
>>> chi = G.character(G.character_table()[Integer(1)])
>>> R = K['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> sorted(G.invariants_of_degree(Integer(2), R=R, chi=chi))
[x*y + (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*x*z
     + (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*y*z,
 x^2 + (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*y^2
     + (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*z^2]

sage: # needs sage.groups sage.rings.number_field
sage: S3 = MatrixGroup(SymmetricGroup(3))
sage: chi = S3.character(S3.character_table()[0])
sage: sorted(S3.invariants_of_degree(5, chi=chi))
[x0^3*x1^2 - x0^2*x1^3 - x0^3*x2^2 + x1^3*x2^2 + x0^2*x2^3 - x1^2*x2^3,
 x0^4*x1 - x0*x1^4 - x0^4*x2 + x1^4*x2 + x0*x2^4 - x1*x2^4]

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> S3 = MatrixGroup(SymmetricGroup(Integer(3)))
>>> chi = S3.character(S3.character_table()[Integer(0)])
>>> sorted(S3.invariants_of_degree(Integer(5), chi=chi))
[x0^3*x1^2 - x0^2*x1^3 - x0^3*x2^2 + x1^3*x2^2 + x0^2*x2^3 - x1^2*x2^3,
 x0^4*x1 - x0*x1^4 - x0^4*x2 + x1^4*x2 + x0*x2^4 - x1*x2^4]
module_composition_factors(algorithm=None)[source]#

Return a list of triples consisting of [base field, dimension, irreducibility], for each of the Meataxe composition factors modules. The algorithm="verbose" option returns more information, but in Meataxe notation.

EXAMPLES:

sage: F = GF(3); MS = MatrixSpace(F,4,4)
sage: M = MS(0)
sage: M[0,1] = 1; M[1,2] = 1; M[2,3] = 1; M[3,0] = 1
sage: G = MatrixGroup([M])
sage: G.module_composition_factors()
[(Finite Field of size 3, 1, True),
 (Finite Field of size 3, 1, True),
 (Finite Field of size 3, 2, True)]
sage: F = GF(7); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[0,1],[-1,0]]), MS([[1,1],[2,3]])]
sage: G = MatrixGroup(gens)
sage: G.module_composition_factors()
[(Finite Field of size 7, 2, True)]

>>> from sage.all import *
>>> F = GF(Integer(3)); MS = MatrixSpace(F,Integer(4),Integer(4))
>>> M = MS(Integer(0))
>>> M[Integer(0),Integer(1)] = Integer(1); M[Integer(1),Integer(2)] = Integer(1); M[Integer(2),Integer(3)] = Integer(1); M[Integer(3),Integer(0)] = Integer(1)
>>> G = MatrixGroup([M])
>>> G.module_composition_factors()
[(Finite Field of size 3, 1, True),
 (Finite Field of size 3, 1, True),
 (Finite Field of size 3, 2, True)]
>>> F = GF(Integer(7)); MS = MatrixSpace(F,Integer(2),Integer(2))
>>> gens = [MS([[Integer(0),Integer(1)],[-Integer(1),Integer(0)]]), MS([[Integer(1),Integer(1)],[Integer(2),Integer(3)]])]
>>> G = MatrixGroup(gens)
>>> G.module_composition_factors()
[(Finite Field of size 7, 2, True)]

Type G.module_composition_factors(algorithm='verbose') to get a more verbose version.

For more on MeatAxe notation, see GAP: chap69.

molien_series(chi=None, return_series=True, prec=20, variable='t')[source]#

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

\[\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\text{det}(I-tg)},\]

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

\[\frac{1}{|G|}\sum_{g \in G} \frac{\chi(g)}{\prod_{i=1}^n (1 - t\omega^{k_i(g)})},\]

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 (possibly infinite, in which case it is computed lazily)

  • 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()              # needs sage.rings.number_field
Traceback (most recent call last):
...
NotImplementedError: characteristic cannot divide group order

>>> from sage.all import *
>>> MatrixGroup(matrix(QQ,Integer(2),Integer(2),[Integer(1),Integer(1),Integer(0),Integer(1)])).molien_series()
Traceback (most recent call last):
...
NotImplementedError: only implemented for finite groups
>>> MatrixGroup(matrix(GF(Integer(3)),Integer(2),Integer(2),[Integer(1),Integer(1),Integer(0),Integer(1)])).molien_series()              # needs sage.rings.number_field
Traceback (most recent call last):
...
NotImplementedError: characteristic cannot divide group order

Tetrahedral Group:

sage: # needs sage.rings.number_field
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)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(4), names=('i',)); (i,) = K._first_ngens(1)
>>> Tetra =  MatrixGroup([(-Integer(1)+i)/Integer(2),(-Integer(1)+i)/Integer(2), (Integer(1)+i)/Integer(2),(-Integer(1)-i)/Integer(2)], [Integer(0),i, -i,Integer(0)])
>>> Tetra.molien_series(prec=Integer(30))
1 + t^8 + 2*t^12 + t^16 + 2*t^20 + 3*t^24 + 2*t^28 + O(t^30)
>>> mol = Tetra.molien_series(return_series=False); mol
(t^8 - t^4 + 1)/(t^16 - t^12 - t^4 + 1)
>>> mol.parent()
Fraction Field of Univariate Polynomial Ring in t over Integer Ring
>>> chi = Tetra.character(Tetra.character_table()[Integer(1)])
>>> Tetra.molien_series(chi, prec=Integer(30), variable='u')
u^6 + u^14 + 2*u^18 + u^22 + 2*u^26 + 3*u^30 + 2*u^34 + O(u^36)
>>> chi = Tetra.character(Tetra.character_table()[Integer(2)])
>>> Tetra.molien_series(chi)
t^10 + t^14 + t^18 + 2*t^22 + 2*t^26 + O(t^30)

sage: # needs sage.groups sage.rings.number_field
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
sage: mol = S3.molien_series(prec=oo); mol
1 + t + 2*t^2 + 3*t^3 + 4*t^4 + 5*t^5 + 7*t^6 + O(t^7)
sage: mol.parent()
Lazy Taylor Series Ring in t over Integer Ring

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> S3 = MatrixGroup(SymmetricGroup(Integer(3)))
>>> mol = S3.molien_series(prec=Integer(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)
>>> mol.parent()
Power Series Ring in t over Integer Ring
>>> mol = S3.molien_series(prec=oo); mol
1 + t + 2*t^2 + 3*t^3 + 4*t^4 + 5*t^5 + 7*t^6 + O(t^7)
>>> mol.parent()
Lazy Taylor Series Ring in t over Integer Ring

Octahedral Group:

sage: # needs sage.rings.number_field
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],          # needs sage.symbolic
....:                    [(1+i)/a, 0,  0, (1-i)/a])
sage: Octa.molien_series(prec=30)                                           # needs sage.symbolic
1 + t^8 + t^12 + t^16 + t^18 + t^20 + 2*t^24 + t^26 + t^28 + O(t^30)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(8), names=('v',)); (v,) = K._first_ngens(1)
>>> a = v - v**Integer(3)  # sqrt(2)
>>> i = v**Integer(2)
>>> Octa = MatrixGroup([(-Integer(1)+i)/Integer(2), (-Integer(1)+i)/Integer(2),  (Integer(1)+i)/Integer(2), (-Integer(1)-i)/Integer(2)],          # needs sage.symbolic
...                    [(Integer(1)+i)/a, Integer(0),  Integer(0), (Integer(1)-i)/a])
>>> Octa.molien_series(prec=Integer(30))                                           # needs sage.symbolic
1 + t^8 + t^12 + t^16 + t^18 + t^20 + 2*t^24 + t^26 + t^28 + O(t^30)

Icosahedral Group:

sage: # needs sage.rings.number_field
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)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(10), names=('v',)); (v,) = K._first_ngens(1)
>>> z5 = v**Integer(2)
>>> i = z5**Integer(5)
>>> a = Integer(2)*z5**Integer(3) + Integer(2)*z5**Integer(2) + Integer(1)  #sqrt(5)
>>> Ico = MatrixGroup([[z5**Integer(3),Integer(0), Integer(0),z5**Integer(2)],
...                    [Integer(0),Integer(1), -Integer(1),Integer(0)],
...                    [(z5**Integer(4)-z5)/a, (z5**Integer(2)-z5**Integer(3))/a,
...                     (z5**Integer(2)-z5**Integer(3))/a, -(z5**Integer(4)-z5)/a]])
>>> Ico.molien_series(prec=Integer(40))
1 + t^12 + t^20 + t^24 + t^30 + t^32 + t^36 + O(t^40)

sage: # needs sage.groups sage.rings.number_field
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)

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> G = MatrixGroup(CyclicPermutationGroup(Integer(3)))
>>> chi = G.character(G.character_table()[Integer(1)])
>>> G.molien_series(chi, prec=Integer(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: # needs sage.groups sage.rings.number_field
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)

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> K = GF(Integer(5))
>>> S = MatrixGroup(SymmetricGroup(Integer(4)))
>>> G = MatrixGroup([matrix(K, Integer(4), Integer(4),
...                         [K(y) for u in m.list() for y in u])
...                  for m in S.gens()])
>>> G.molien_series(return_series=False)
1/(t^10 - t^9 - t^8 + 2*t^5 - t^2 - t + 1)

sage: # needs sage.rings.number_field
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)

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> i = GF(Integer(7))(Integer(3))
>>> G = MatrixGroup([[i**Integer(3),Integer(0), Integer(0),-i**Integer(3)], [i**Integer(2),Integer(0), Integer(0),-i**Integer(2)]])
>>> chi = G.character(G.character_table()[Integer(4)])
>>> G.molien_series(chi)
3*t^5 + 6*t^11 + 9*t^17 + 12*t^23 + O(t^25)
reynolds_operator(poly, chi=None)[source]#

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

be the ring of invariants of \(G\) relative to \(\chi\). Then the Reynolds operator is a map \(R\) from \(K[x]\) into \(K[x]^G_{\chi}\) defined by

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)                                               # needs sage.rings.number_field
1/3*x^3*y*z + 1/3*x*y^3*z + 1/3*x*y*z^3

>>> from sage.all import *
>>> S3 = MatrixGroup(SymmetricGroup(Integer(3)))
>>> R = QQ['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> f = x*y*z**Integer(3)
>>> S3.reynolds_operator(f)                                               # needs sage.rings.number_field
1/3*x^3*y*z + 1/3*x*y^3*z + 1/3*x*y*z^3

sage: # needs sage.groups sage.rings.number_field
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

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> G = MatrixGroup(CyclicPermutationGroup(Integer(4)))
>>> chi = G.character(G.character_table()[Integer(3)])
>>> K = CyclotomicField(Integer(4), names=('v',)); (v,) = K._first_ngens(1)
>>> R = K['x, y, z, w']; (x, y, z, w,) = R._first_ngens(4)
>>> G.reynolds_operator(x, chi)
1/4*x + (1/4*v)*y - 1/4*z + (-1/4*v)*w
>>> chi = G.character(G.character_table()[Integer(2)])
>>> R = QQ['x, y, z, w']; (x, y, z, w,) = R._first_ngens(4)
>>> 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: # needs sage.groups sage.rings.number_field
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 - 4/3)*x^5*z
          + (2/3*izeta3^3 + izeta3^2 + 8/3*izeta3 + 1)*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

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> K = CyclotomicField(Integer(4), names=('i',)); (i,) = K._first_ngens(1)
>>> G =  MatrixGroup(CyclicPermutationGroup(Integer(3)))
>>> chi = G.character(G.character_table()[Integer(1)])
>>> R = K['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> G.reynolds_operator(x*y**Integer(5), chi)
1/3*x*y^5 + (-2/3*izeta3^3 - izeta3^2 - 8/3*izeta3 - 4/3)*x^5*z
          + (2/3*izeta3^3 + izeta3^2 + 8/3*izeta3 + 1)*y*z^5
>>> R = QQbar['x, y, z']; (x, y, z,) = R._first_ngens(3)
>>> G.reynolds_operator(x*y**Integer(5), chi)
 1/3*x*y^5 + (-0.1666666666666667? + 0.2886751345948129?*I)*x^5*z
           + (-0.1666666666666667? - 0.2886751345948129?*I)*y*z^5

sage: # needs sage.rings.number_field
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

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> K = CyclotomicField(Integer(4), names=('i',)); (i,) = K._first_ngens(1)
>>> Tetra =  MatrixGroup([(-Integer(1)+i)/Integer(2),(-Integer(1)+i)/Integer(2), (Integer(1)+i)/Integer(2),(-Integer(1)-i)/Integer(2)], [Integer(0),i, -i,Integer(0)])
>>> chi = Tetra.character(Tetra.character_table()[Integer(4)])
>>> L = QuadraticField(-Integer(3), names=('v',)); (v,) = L._first_ngens(1)
>>> R = L['x, y']; (x, y,) = R._first_ngens(2)
>>> Tetra.reynolds_operator(x**Integer(4))
0
>>> Tetra.reynolds_operator(x**Integer(4), chi)
1/4*x^4 + (1/2*v)*x^2*y^2 + 1/4*y^4
>>> R = L['x']; (x,) = R._first_ngens(1)
>>> LL = L.extension(x**Integer(2) + v, names=('w',)); (w,) = LL._first_ngens(1)
>>> R = LL['x, y']; (x, y,) = R._first_ngens(2)
>>> Tetra.reynolds_operator(x**Integer(4), chi)
Traceback (most recent call last):
...
NotImplementedError: only implemented for absolute fields

sage: # needs sage.groups sage.rings.number_field
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

>>> from sage.all import *
>>> # needs sage.groups sage.rings.number_field
>>> G =  MatrixGroup(DihedralGroup(Integer(4)))
>>> chi = G.character(G.character_table()[Integer(1)])
>>> R = QQ['x, y']; (x, y,) = R._first_ngens(2)
>>> f = x**Integer(4)
>>> G.reynolds_operator(f, chi)
Traceback (most recent call last):
...
TypeError: number of variables in polynomial must match size of matrices
>>> R = QQ['x, y, z, w']; (x, y, z, w,) = R._first_ngens(4)
>>> f = x**Integer(3)*y
>>> 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

>>> from sage.all import *
>>> G = MatrixGroup([[Integer(0),Integer(1), Integer(1),Integer(0)]])
>>> R = GF(Integer(2))['w, x']; (w, x,) = R._first_ngens(2)
>>> 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])                             # needs sage.rings.number_field
sage: R.<w,x> = GF(7)[]
sage: f = w^5*x + x^6
sage: G.reynolds_operator(f, chi)                                           # needs sage.rings.number_field
Traceback (most recent call last):
...
NotImplementedError: nontrivial characters not implemented for characteristic > 0
sage: G.reynolds_operator(f)
x^6

>>> from sage.all import *
>>> i = GF(Integer(7))(Integer(3))
>>> G = MatrixGroup([[i**Integer(3),Integer(0), Integer(0),-i**Integer(3)], [i**Integer(2),Integer(0), Integer(0),-i**Integer(2)]])
>>> chi = G.character(G.character_table()[Integer(4)])                             # needs sage.rings.number_field
>>> R = GF(Integer(7))['w, x']; (w, x,) = R._first_ngens(2)
>>> f = w**Integer(5)*x + x**Integer(6)
>>> G.reynolds_operator(f, chi)                                           # needs sage.rings.number_field
Traceback (most recent call last):
...
NotImplementedError: nontrivial characters not implemented for characteristic > 0
>>> G.reynolds_operator(f)
x^6

sage: # needs sage.rings.finite_rings
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

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(3)**Integer(2),'t')
>>> G = MatrixGroup([matrix(K, Integer(2), Integer(2), [Integer(0),K.gen(), Integer(1),Integer(0)])])
>>> R = GF(Integer(3))['x, y']; (x, y,) = R._first_ngens(2)
>>> G.reynolds_operator(x**Integer(8))
-x^8 - y^8

sage: # needs sage.rings.finite_rings
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

>>> from sage.all import *
>>> # needs sage.rings.finite_rings
>>> K = GF(Integer(3)**Integer(2),'t')
>>> G = MatrixGroup([matrix(GF(Integer(3)), Integer(2), Integer(2), [Integer(0),Integer(1), Integer(1),Integer(0)])])
>>> R = K['x, y']; (x, y,) = R._first_ngens(2)
>>> f = -K.gen()*x
>>> G.reynolds_operator(f)
t*x + t*y