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finitely_generated.py
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finitely_generated.py
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"""
Finitely Generated Matrix Groups
This class is designed for computing with matrix groups defined by a
finite set of generating matrices.
EXAMPLES::
sage: F = GF(3)
sage: gens = [matrix(F,2, [1,0, -1,1]), matrix(F,2, [1,1,0,1])]
sage: G = MatrixGroup(gens)
sage: G.conjugacy_classes_representatives()
(
[1 0] [0 2] [0 1] [2 0] [0 2] [0 1] [0 2]
[0 1], [1 1], [2 1], [0 2], [1 2], [2 2], [1 0]
)
The finitely generated matrix groups can also be constructed as
subgroups of matrix groups::
sage: SL2Z = SL(2,ZZ)
sage: S, T = SL2Z.gens()
sage: SL2Z.subgroup([T^2])
Subgroup with 1 generators (
[1 2]
[0 1]
) of Special Linear Group of degree 2 over Integer Ring
AUTHORS:
- William Stein: initial version
- David Joyner (2006-03-15): degree, base_ring, _contains_, list,
random, order methods; examples
- William Stein (2006-12): rewrite
- David Joyner (2007-12): Added invariant_generators (with Martin
Albrecht and Simon King)
- David Joyner (2008-08): Added module_composition_factors (interface
to GAP's MeatAxe implementation) and as_permutation_group (returns
isomorphic PermutationGroup).
- Simon King (2010-05): Improve invariant_generators by using GAP
for the construction of the Reynolds operator in Singular.
- Volker Braun (2013-1) port to new Parent, libGAP.
- Sebastian Oehms (2018-07): Added _permutation_group_element_ (Trac #25706)
- Sebastian Oehms (2019-01): Revision of :trac:`25706` (:trac:`26903` and :trac:`27143`).
"""
##############################################################################
# Copyright (C) 2006 David Joyner and William Stein <wstein@gmail.com>
# Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# The full text of the GPL is available at:
#
# https://www.gnu.org/licenses/
##############################################################################
from sage.rings.integer_ring import ZZ
from sage.rings.all import QQbar
from sage.structure.element import is_Matrix
from sage.matrix.matrix_space import MatrixSpace, is_MatrixSpace
from sage.matrix.constructor import matrix
from sage.structure.sequence import Sequence
from sage.misc.cachefunc import cached_method
from sage.modules.free_module_element import vector
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.rings.power_series_ring import PowerSeriesRing
from sage.rings.fraction_field import FractionField
from sage.misc.functional import cyclotomic_polynomial
from sage.rings.number_field.number_field import CyclotomicField
from sage.combinat.integer_vector import IntegerVectors
from sage.groups.matrix_gps.matrix_group import (MatrixGroup_generic,
MatrixGroup_gap)
from sage.groups.matrix_gps.group_element import is_MatrixGroupElement
def normalize_square_matrices(matrices):
"""
Find a common space for all matrices.
OUTPUT:
A list of matrices, all elements of the same matrix space.
EXAMPLES::
sage: from sage.groups.matrix_gps.finitely_generated import normalize_square_matrices
sage: m1 = [[1,2],[3,4]]
sage: m2 = [2, 3, 4, 5]
sage: m3 = matrix(QQ, [[1/2,1/3],[1/4,1/5]])
sage: m4 = MatrixGroup(m3).gen(0)
sage: normalize_square_matrices([m1, m2, m3, m4])
[
[1 2] [2 3] [1/2 1/3] [1/2 1/3]
[3 4], [4 5], [1/4 1/5], [1/4 1/5]
]
"""
deg = []
gens = []
for m in matrices:
if is_MatrixGroupElement(m):
deg.append(m.parent().degree())
gens.append(m.matrix())
continue
if is_Matrix(m):
if not m.is_square():
raise TypeError('matrix must be square')
deg.append(m.ncols())
gens.append(m)
continue
try:
m = list(m)
except TypeError:
gens.append(m)
continue
if isinstance(m[0], (list, tuple)):
m = [list(_) for _ in m]
degree = ZZ(len(m))
else:
degree, rem = ZZ(len(m)).sqrtrem()
if rem != 0:
raise ValueError('list of plain numbers must have square integer length')
deg.append(degree)
gens.append(matrix(degree, degree, m))
deg = set(deg)
if len(set(deg)) != 1:
raise ValueError('not all matrices have the same size')
gens = Sequence(gens, immutable=True)
MS = gens.universe()
if not is_MatrixSpace(MS):
raise TypeError('all generators must be matrices')
if MS.nrows() != MS.ncols():
raise ValueError('matrices must be square')
return gens
def QuaternionMatrixGroupGF3():
r"""
The quaternion group as a set of `2\times 2` matrices over `GF(3)`.
OUTPUT:
A matrix group consisting of `2\times 2` matrices with
elements from the finite field of order 3. The group is
the quaternion group, the nonabelian group of order 8 that
is not isomorphic to the group of symmetries of a square
(the dihedral group `D_4`).
.. note::
This group is most easily available via ``groups.matrix.QuaternionGF3()``.
EXAMPLES:
The generators are the matrix representations of the
elements commonly called `I` and `J`, while `K`
is the product of `I` and `J`. ::
sage: from sage.groups.matrix_gps.finitely_generated import QuaternionMatrixGroupGF3
sage: Q = QuaternionMatrixGroupGF3()
sage: Q.order()
8
sage: aye = Q.gens()[0]; aye
[1 1]
[1 2]
sage: jay = Q.gens()[1]; jay
[2 1]
[1 1]
sage: kay = aye*jay; kay
[0 2]
[1 0]
TESTS::
sage: groups.matrix.QuaternionGF3()
Matrix group over Finite Field of size 3 with 2 generators (
[1 1] [2 1]
[1 2], [1 1]
)
sage: Q = QuaternionMatrixGroupGF3()
sage: QP = Q.as_permutation_group()
sage: QP.is_isomorphic(QuaternionGroup())
True
sage: H = DihedralGroup(4)
sage: H.order()
8
sage: QP.is_abelian(), H.is_abelian()
(False, False)
sage: QP.is_isomorphic(H)
False
"""
from sage.rings.finite_rings.finite_field_constructor import FiniteField
from sage.matrix.matrix_space import MatrixSpace
MS = MatrixSpace(FiniteField(3), 2)
aye = MS([1,1,1,2])
jay = MS([2,1,1,1])
return MatrixGroup([aye, jay])
def MatrixGroup(*gens, **kwds):
r"""
Return the matrix group with given generators.
INPUT:
- ``*gens`` -- matrices, or a single list/tuple/iterable of
matrices, or a matrix group.
- ``check`` -- boolean keyword argument (optional, default:
``True``). Whether to check that each matrix is invertible.
EXAMPLES::
sage: F = GF(5)
sage: gens = [matrix(F,2,[1,2, -1, 1]), matrix(F,2, [1,1, 0,1])]
sage: G = MatrixGroup(gens); G
Matrix group over Finite Field of size 5 with 2 generators (
[1 2] [1 1]
[4 1], [0 1]
)
In the second example, the generators are a matrix over
`\ZZ`, a matrix over a finite field, and the integer
`2`. Sage determines that they both canonically map to
matrices over the finite field, so creates that matrix group
there::
sage: gens = [matrix(2,[1,2, -1, 1]), matrix(GF(7), 2, [1,1, 0,1]), 2]
sage: G = MatrixGroup(gens); G
Matrix group over Finite Field of size 7 with 3 generators (
[1 2] [1 1] [2 0]
[6 1], [0 1], [0 2]
)
Each generator must be invertible::
sage: G = MatrixGroup([matrix(ZZ,2,[1,2,3,4])])
Traceback (most recent call last):
...
ValueError: each generator must be an invertible matrix
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: MatrixGroup([MS.0])
Traceback (most recent call last):
...
ValueError: each generator must be an invertible matrix
sage: MatrixGroup([MS.0], check=False) # works formally but is mathematical nonsense
Matrix group over Finite Field of size 5 with 1 generators (
[1 0]
[0 0]
)
Some groups are not supported, or do not have much functionality
implemented::
sage: G = SL(0, QQ)
Traceback (most recent call last):
...
ValueError: the degree must be at least 1
sage: SL2C = SL(2, CC); SL2C
Special Linear Group of degree 2 over Complex Field with 53 bits of precision
sage: SL2C.gens()
Traceback (most recent call last):
...
AttributeError: 'LinearMatrixGroup_generic_with_category' object has no attribute 'gens'
"""
if isinstance(gens[-1], dict): # hack for unpickling
kwds.update(gens[-1])
gens = gens[:-1]
check = kwds.get('check', True)
if len(gens) == 1:
if isinstance(gens[0], (list, tuple)):
gens = list(gens[0])
else:
try:
gens = [g.matrix() for g in gens[0]]
except AttributeError:
pass
if len(gens) == 0:
raise ValueError('need at least one generator')
gens = normalize_square_matrices(gens)
if check and any(not g.is_invertible() for g in gens):
raise ValueError('each generator must be an invertible matrix')
MS = gens.universe()
base_ring = MS.base_ring()
degree = ZZ(MS.ncols()) # == MS.nrows()
from sage.libs.gap.libgap import libgap
category = kwds.get('category', None)
try:
gap_gens = [libgap(matrix_gen) for matrix_gen in gens]
gap_group = libgap.Group(gap_gens)
return FinitelyGeneratedMatrixGroup_gap(degree, base_ring, gap_group,
category=category)
except (TypeError, ValueError):
return FinitelyGeneratedMatrixGroup_generic(degree, base_ring, gens,
category=category)
###################################################################
#
# Matrix group over a generic ring
#
###################################################################
class FinitelyGeneratedMatrixGroup_generic(MatrixGroup_generic):
"""
TESTS::
sage: m1 = matrix(SR, [[1,2],[3,4]])
sage: m2 = matrix(SR, [[1,3],[-1,0]])
sage: MatrixGroup(m1) == MatrixGroup(m1)
True
sage: MatrixGroup(m1) == MatrixGroup(m1.change_ring(QQ))
False
sage: MatrixGroup(m1) == MatrixGroup(m2)
False
sage: MatrixGroup(m1, m2) == MatrixGroup(m2, m1)
False
sage: m1 = matrix(QQ, [[1,2],[3,4]])
sage: m2 = matrix(QQ, [[1,3],[-1,0]])
sage: MatrixGroup(m1) == MatrixGroup(m1)
True
sage: MatrixGroup(m1) == MatrixGroup(m2)
False
sage: MatrixGroup(m1, m2) == MatrixGroup(m2, m1)
False
sage: G = GL(2, GF(3))
sage: H = G.as_matrix_group()
sage: H == G, G == H
(True, True)
"""
def __init__(self, degree, base_ring, generator_matrices, category=None):
"""
Matrix group generated by a finite number of matrices.
EXAMPLES::
sage: m1 = matrix(SR, [[1,2],[3,4]])
sage: m2 = matrix(SR, [[1,3],[-1,0]])
sage: G = MatrixGroup(m1, m2)
sage: TestSuite(G).run()
sage: type(G)
<class 'sage.groups.matrix_gps.finitely_generated.FinitelyGeneratedMatrixGroup_generic_with_category'>
sage: from sage.groups.matrix_gps.finitely_generated import \
....: FinitelyGeneratedMatrixGroup_generic
sage: G = FinitelyGeneratedMatrixGroup_generic(2, QQ, [matrix(QQ,[[1,2],[3,4]])])
sage: G.gens()
(
[1 2]
[3 4]
)
"""
self._gens_matrix = generator_matrices
MatrixGroup_generic.__init__(self, degree, base_ring, category=category)
@cached_method
def gens(self):
"""
Return the generators of the matrix group.
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]), MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: gens[0] in G
True
sage: gens = G.gens()
sage: gens[0] in G
True
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: F = GF(5); MS = MatrixSpace(F,2,2)
sage: G = MatrixGroup([MS(1), MS([1,2,3,4])])
sage: G
Matrix group over Finite Field of size 5 with 2 generators (
[1 0] [1 2]
[0 1], [3 4]
)
sage: G.gens()
(
[1 0] [1 2]
[0 1], [3 4]
)
"""
return tuple(self.element_class(self, x, check=False, convert=False)
for x in self._gens_matrix)
def gen(self, i):
"""
Return the `i`-th generator
OUTPUT:
The `i`-th generator of the group.
EXAMPLES::
sage: H = GL(2, GF(3))
sage: h1, h2 = H([[1,0],[2,1]]), H([[1,1],[0,1]])
sage: G = H.subgroup([h1, h2])
sage: G.gen(0)
[1 0]
[2 1]
sage: G.gen(0).matrix() == h1.matrix()
True
"""
return self.gens()[i]
def ngens(self):
"""
Return the number of generators
OUTPUT:
An integer. The number of generators.
EXAMPLES::
sage: H = GL(2, GF(3))
sage: h1, h2 = H([[1,0],[2,1]]), H([[1,1],[0,1]])
sage: G = H.subgroup([h1, h2])
sage: G.ngens()
2
"""
return len(self._gens_matrix)
def __reduce__(self):
"""
Used for pickling.
TESTS::
sage: G = MatrixGroup([matrix(CC, [[1,2],[3,4]]),
....: matrix(CC, [[1,3],[-1,0]])])
sage: loads(dumps(G)) == G
True
Check that :trac:`22128` is fixed::
sage: R = MatrixSpace(SR, 2)
sage: G = MatrixGroup([R([[1, 1], [0, 1]])])
sage: G.register_embedding(R)
sage: loads(dumps(G))
Matrix group over Symbolic Ring with 1 generators (
[1 1]
[0 1]
)
"""
return MatrixGroup, (self._gens_matrix, {'check': False})
def _test_matrix_generators(self, **options):
"""
EXAMPLES::
sage: m1 = matrix(SR, [[1,2],[3,4]])
sage: m2 = matrix(SR, [[1,3],[-1,0]])
sage: G = MatrixGroup(m1, m2)
sage: G._test_matrix_generators()
"""
tester = self._tester(**options)
for g,h in zip(self.gens(), MatrixGroup(self.gens()).gens()):
tester.assertEqual(g.matrix(), h.matrix())
###################################################################
#
# Matrix group over a ring that GAP understands
#
###################################################################
class FinitelyGeneratedMatrixGroup_gap(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.FinitelyGeneratedMatrixGroup_gap_with_category'>
sage: TestSuite(G).run()
"""
def __reduce__(self):
"""
Implement pickling.
EXAMPLES::
sage: m1 = matrix(QQ, [[1,2],[3,4]])
sage: m2 = matrix(QQ, [[1,3],[-1,0]])
sage: loads(MatrixGroup(m1, m2).dumps())
Matrix group over Rational Field with 2 generators (
[1 2] [ 1 3]
[3 4], [-1 0]
)
"""
return (MatrixGroup,
tuple(g.matrix() for g in self.gens()) + ({'check':False},))
def as_permutation_group(self, algorithm=None, seed=None):
r"""
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
A finite subgroup of GL(12,Z) 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 == Psmaller # see the note below
True
sage: Psmaller = G.as_permutation_group(algorithm="smaller")
sage: P == Psmaller
False
sage: P.cardinality()
21499084800
sage: P.degree()
144
sage: Psmaller.cardinality()
21499084800
sage: Psmaller.degree()
80
.. 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.
TESTS::
sage: A= matrix(QQ, 2, [0, 1, 1, 0])
sage: B= matrix(QQ, 2, [1, 0, 0, 1])
sage: a, b= MatrixGroup([A, B]).as_permutation_group().gens()
sage: a.order(), b.order()
(2, 1)
The above example in GL(12,Z), reduced modulo 7::
sage: MS = MatrixSpace(GF(7), 12, 12)
sage: G = MatrixGroup([MS(g) for g in GG.GeneratorsOfGroup()])
sage: G.cardinality()
21499084800
sage: P = G.as_permutation_group()
sage: P.cardinality()
21499084800
Check that large degree is still working::
sage: Sp(6,3).as_permutation_group().cardinality()
9170703360
Check that :trac:`25706` still works after :trac:`26903`::
sage: MG = GU(3,2).as_matrix_group()
sage: PG = MG.as_permutation_group()
sage: mg = MG.an_element()
sage: PG(mg).order() # particular element depends on the set of GAP packages installed
6
"""
# Note that the output of IsomorphismPermGroup() depends on
# memory locations and will change if you change the order of
# doctests and/or architecture
from sage.groups.perm_gps.permgroup import PermutationGroup
if not self.is_finite():
raise NotImplementedError("group must be finite")
if seed is not None:
from sage.libs.gap.libgap import libgap
libgap.set_seed(ZZ(seed))
iso = self._libgap_().IsomorphismPermGroup()
if algorithm == "smaller":
iso = iso.Image().SmallerDegreePermutationRepresentation()
return PermutationGroup(iso.Image().GeneratorsOfGroup().sage(),
canonicalize=False)
def module_composition_factors(self, algorithm=None):
r"""
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)]
Type ``G.module_composition_factors(algorithm='verbose')`` to get a
more verbose version.
For more on MeatAxe notation, see
https://www.gap-system.org/Manuals/doc/ref/chap69.html
"""
from sage.libs.gap.libgap import libgap
F = self.base_ring()
if not F.is_finite():
raise NotImplementedError("base ring must be finite")
n = self.degree()
MS = MatrixSpace(F, n, n)
mats = [MS(g.matrix()) for g in self.gens()]
# initializing list of mats by which the gens act on self
mats_gap = libgap(mats)
M = mats_gap.GModuleByMats(F)
compo = libgap.function_factory('MTX.CompositionFactors')
MCFs = compo(M)
if algorithm == "verbose":
print(str(MCFs) + "\n")
return sorted((MCF['field'].sage(),
MCF['dimension'].sage(),
MCF['IsIrreducible'].sage()) for MCF in MCFs)
def invariant_generators(self, ring=None):
r"""
Return invariant ring generators.
INPUT:
- ``ring`` -- (default: None) The polynomial ring where the result will live.
If it is not given, it will be created according to the group.
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()
[x0^7*x1 - x0*x1^7,
x0^12 - 2*x0^9*x1^3 - x0^6*x1^6 + 2*x0^3*x1^9 + x1^12,
x0^18 + 2*x0^15*x1^3 + 3*x0^12*x1^6 + 3*x0^6*x1^12 - 2*x0^3*x1^15 + x1^18]
sage: q = 4; a = 2
sage: MS = MatrixSpace(QQ, 2, 2)
sage: gen1 = [[1/a,(q-1)/a],[1/a, -1/a]]; 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()
[x0^2 + 3*x1^2, x0^6 + 15*x0^4*x1^2 + 15*x0^2*x1^4 + 33*x1^6]
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: R = PolynomialRing(F, 'x,y')
sage: G.invariant_generators(R)
[x^4 + 2*x^2*y^2 + y^4,
x^5*y - x*y^5,
x^8 + 28/9*x^6*y^2 + 70/9*x^4*y^4 + 28/9*x^2*y^6 + y^8]
An example of the modular case::
sage: F = FiniteField(2)
sage: m1 = matrix(F, [[0,1],[1,0]])
sage: G = MatrixGroup([m1])
sage: G.invariant_generators()
[x0 + x1, x0*x1, 1]
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`.
"""
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.modules.free_module_element import vector
if not ring:
F = self.base_ring()
if not F.is_field():
F = F.fraction_field()
n = self.matrix_space().nrows()
R = PolynomialRing(F, n, ['x{}'.format(i) for i in range(n)])
else:
R = ring
char = R.base_ring().characteristic()
import sage.libs.singular.function_factory
from sage.libs.singular.function import singular_function
sage.libs.singular.function_factory.lib('finvar.lib')
if char == 0 or self.cardinality() % char != 0: #non modular case
inrey = singular_function('invariant_algebra_reynolds')
L = matrix([a.matrix()*vector(R.gens()) for a in self.list()])
invars = inrey(L, 0)
return [R(a) for a in invars[0]]
else: #modular case
primcharp = singular_function('primary_charp_without')
secnm = singular_function('secondary_not_cohen_macaulay')
lgens = [a.matrix().change_ring(R) for a in self.gens()] + [0]
primgens = primcharp(*lgens)
secgens = secnm(primgens, *lgens)
return [R(a) for a in primgens.list() + secgens.list()]
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\}