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sage.groups.matrix_gps: Docstring cosmetics
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@mkoeppe
mkoeppe committed Apr 14, 2023
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12 changes: 3 additions & 9 deletions src/sage/groups/affine_gps/affine_group.py
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Expand Up @@ -301,9 +301,7 @@ def degree(self):
"""
Return the dimension of the affine space.
OUTPUT:
An integer.
OUTPUT: An integer.
EXAMPLES::
Expand Down Expand Up @@ -388,9 +386,7 @@ def linear(self, A):
- ``A`` -- anything that determines a matrix
OUTPUT:
The affine group element `x \mapsto A x`.
OUTPUT: The affine group element `x \mapsto A x`.
EXAMPLES::
Expand All @@ -411,9 +407,7 @@ def translation(self, b):
- ``b`` -- anything that determines a vector
OUTPUT:
The affine group element `x \mapsto x + b`.
OUTPUT: The affine group element `x \mapsto x + b`.
EXAMPLES::
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2 changes: 1 addition & 1 deletion src/sage/groups/affine_gps/euclidean_group.py
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Expand Up @@ -21,7 +21,7 @@

class EuclideanGroup(AffineGroup):
r"""
an Euclidean group.
A Euclidean group.
The Euclidean group `E(A)` (or general affine group) of an affine
space `A` is the group of all invertible affine transformations from
Expand Down
26 changes: 7 additions & 19 deletions src/sage/groups/affine_gps/group_element.py
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Expand Up @@ -67,16 +67,14 @@ class AffineGroupElement(MultiplicativeGroupElement):
correct vector space.
- ``check`` - bool (default: ``True``). Whether to do some
checks or just accept the input as valid.
checks or just accept the input as valid.
As a special case, ``A`` can be a matrix obtained from
:meth:`matrix`, that is, one row and one column larger. In
that case, the group element defining that matrix is
reconstructed.
OUTPUT:
The affine group element `x \mapsto Ax + b`
OUTPUT: The affine group element `x \mapsto Ax + b`
EXAMPLES::
Expand Down Expand Up @@ -144,9 +142,7 @@ def A(self):
"""
Return the general linear part of an affine group element.
OUTPUT:
The matrix `A` of the affine group element `Ax + b`.
OUTPUT: The matrix `A` of the affine group element `Ax + b`.
EXAMPLES::
Expand All @@ -163,9 +159,7 @@ def b(self):
"""
Return the translation part of an affine group element.
OUTPUT:
The vector `b` of the affine group element `Ax + b`.
OUTPUT: The vector `b` of the affine group element `Ax + b`.
EXAMPLES::
Expand Down Expand Up @@ -361,9 +355,7 @@ def __call__(self, v):
- ``v`` -- a polynomial, a multivariate polynomial, a polyhedron, a
vector, or anything that can be converted into a vector.
OUTPUT:
The image of ``v`` under the affine group element.
OUTPUT: The image of ``v`` under the affine group element.
EXAMPLES::
Expand Down Expand Up @@ -459,9 +451,7 @@ def __invert__(self):
"""
Return the inverse group element.
OUTPUT:
Another affine group element.
OUTPUT: Another affine group element.
EXAMPLES::
Expand All @@ -488,9 +478,7 @@ def _richcmp_(self, other, op):
"""
Compare ``self`` with ``other``.
OUTPUT:
boolean
OUTPUT: boolean
EXAMPLES::
Expand Down
12 changes: 7 additions & 5 deletions src/sage/groups/matrix_gps/coxeter_group.py
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Expand Up @@ -186,10 +186,12 @@ class CoxeterMatrixGroup(UniqueRepresentation, FinitelyGeneratedMatrixGroup_gene
[2 2 3 2 1]
sage: W = CoxeterGroup(['H',3], implementation="reflection")
sage: W
Finite Coxeter group over Number Field in a with defining polynomial x^2 - 5 with a = 2.236067977499790? with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
Finite Coxeter group over
Number Field in a with defining polynomial x^2 - 5 with a = 2.236067977499790?
with Coxeter matrix:
[1 3 2]
[3 1 5]
[2 5 1]
"""
@staticmethod
def __classcall_private__(cls, data, base_ring=None, index_set=None):
Expand Down Expand Up @@ -834,7 +836,7 @@ def action_on_root_indices(self, i, side="left"):
"""
Return the action on the set of roots.
The roots are ordered as in the output of the method `roots`.
The roots are ordered as in the output of the method :meth:`roots`.
EXAMPLES::
Expand Down
2 changes: 1 addition & 1 deletion src/sage/groups/matrix_gps/finitely_generated.py
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Expand Up @@ -141,7 +141,7 @@ def normalize_square_matrices(matrices):

def QuaternionMatrixGroupGF3():
r"""
The quaternion group as a set of `2\times 2` matrices over `GF(3)`.
The quaternion group as a set of `2\times 2` matrices over `\GF{3}`.
OUTPUT:
Expand Down
53 changes: 31 additions & 22 deletions src/sage/groups/matrix_gps/finitely_generated_gap.py
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Expand Up @@ -93,7 +93,7 @@ def as_permutation_group(self, algorithm=None, seed=None):
- ``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.
option, since GAP produces random output in that context.
OUTPUT:
Expand All @@ -112,7 +112,7 @@ def as_permutation_group(self, algorithm=None, seed=None):
sage: G.as_permutation_group().order()
2
A finite subgroup of GL(12,Z) as a permutation group::
A finite subgroup of `GL(12,\ZZ)` as a permutation group::
sage: imf = libgap.function_factory('ImfMatrixGroup')
sage: GG = imf( 12, 3 )
Expand All @@ -137,7 +137,7 @@ def as_permutation_group(self, algorithm=None, seed=None):
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
to construct the :class:`MatrixGroup` `G` over `\GF{7}. The section
"Irreducible Maximal Finite Integral Matrix Groups" in the
GAP reference manual has more details.
Expand All @@ -148,7 +148,7 @@ def as_permutation_group(self, algorithm=None, seed=None):
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."
``SmallerDegreePermutationRepresentation``."
To obtain a reproducible result the optional argument ``seed``
may be used as in the example above.
Expand All @@ -161,7 +161,7 @@ def as_permutation_group(self, algorithm=None, seed=None):
sage: a.order(), b.order()
(2, 1)
The above example in GL(12,Z), reduced modulo 7::
The above example in `GL(12,\ZZ)`, reduced modulo 7::
sage: MS = MatrixSpace(GF(7), 12, 12)
sage: G = MatrixGroup([MS(g) for g in GG.GeneratorsOfGroup()])
Expand Down Expand Up @@ -481,7 +481,8 @@ def molien_series(self, chi=None, return_series=True, prec=20, variable='t'):
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 = 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)
Expand All @@ -491,7 +492,9 @@ def molien_series(self, chi=None, return_series=True, prec=20, variable='t'):
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 = 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)
Expand All @@ -500,20 +503,22 @@ def molien_series(self, chi=None, return_series=True, prec=20, variable='t'):
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)
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 = 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: 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)
Expand Down Expand Up @@ -597,7 +602,7 @@ def reynolds_operator(self, poly, chi=None):
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
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
.. MATH:
Expand Down Expand Up @@ -644,10 +649,12 @@ def reynolds_operator(self, poly, chi=None):
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
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
1/3*x*y^5 + (-0.1666666666666667? + 0.2886751345948129?*I)*x^5*z
+ (-0.1666666666666667? - 0.2886751345948129?*I)*y*z^5
::
Expand All @@ -661,7 +668,7 @@ def reynolds_operator(self, poly, chi=None):
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: LL.<w> = L.extension(x^2 + v)
sage: R.<x,y> = LL[]
sage: Tetra.reynolds_operator(x^4, chi)
Traceback (most recent call last):
Expand All @@ -684,9 +691,9 @@ def reynolds_operator(self, poly, chi=None):
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::
Characteristic `p>0` examples::
sage: G = MatrixGroup([[0,1,1,0]])
sage: G = MatrixGroup([[0,1, 1,0]])
sage: R.<w,x> = GF(2)[]
sage: G.reynolds_operator(x)
Traceback (most recent call last):
Expand All @@ -696,7 +703,7 @@ def reynolds_operator(self, poly, chi=None):
::
sage: i = GF(7)(3)
sage: G = MatrixGroup([[i^3,0,0,-i^3],[i^2,0,0,-i^2]])
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
Expand All @@ -710,15 +717,15 @@ def reynolds_operator(self, poly, chi=None):
::
sage: K = GF(3^2,'t')
sage: G = MatrixGroup([matrix(K,2,2, [0,K.gen(),1,0])])
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: 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)
Expand Down Expand Up @@ -863,16 +870,18 @@ def invariants_of_degree(self, deg, chi=None, R=None):
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]
[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: 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]
x0^4*x1 - x0*x1^4 - x0^4*x2 + x1^4*x2 + x0*x2^4 - x1*x2^4]
"""
D = self.degree()
deg = int(deg)
Expand Down

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