Elements of a semimonomial transformation group#

The semimonomial transformation group of degree \(n\) over a ring \(R\) is the semidirect product of the monomial transformation group of degree \(n\) (also known as the complete monomial group over the group of units \(R^{\times}\) of \(R\)) and the group of ring automorphisms.

The multiplication of two elements \((\phi, \pi, \alpha)(\psi, \sigma, \beta)\) with

  • \(\phi, \psi \in {R^{\times}}^n\)

  • \(\pi, \sigma \in S_n\) (with the multiplication \(\pi\sigma\) done from left to right (like in GAP) – that is, \((\pi\sigma)(i) = \sigma(\pi(i))\) for all \(i\).)

  • \(\alpha, \beta \in Aut(R)\)

is defined by

\[(\phi, \pi, \alpha)(\psi, \sigma, \beta) = (\phi \cdot \psi^{\pi, \alpha}, \pi\sigma, \alpha \circ \beta)\]

with \(\psi^{\pi, \alpha} = (\alpha(\psi_{\pi(1)-1}), \ldots, \alpha(\psi_{\pi(n)-1}))\) and an elementwisely defined multiplication of vectors. (The indexing of vectors is \(0\)-based here, so \(\psi = (\psi_0, \psi_1, \ldots, \psi_{n-1})\).)

The parent is SemimonomialTransformationGroup.

AUTHORS:

  • Thomas Feulner (2012-11-15): initial version

  • Thomas Feulner (2013-12-27): Issue #15576 dissolve dependency on

    Permutations.options.mul

EXAMPLES:

sage: S = SemimonomialTransformationGroup(GF(4, 'a'), 4)
sage: G = S.gens()
sage: G[0]*G[1]
((a, 1, 1, 1); (1,2,3,4), Ring endomorphism of Finite Field in a of size 2^2
  Defn: a |--> a)

>>> from sage.all import *
>>> S = SemimonomialTransformationGroup(GF(Integer(4), 'a'), Integer(4))
>>> G = S.gens()
>>> G[Integer(0)]*G[Integer(1)]
((a, 1, 1, 1); (1,2,3,4), Ring endomorphism of Finite Field in a of size 2^2
  Defn: a |--> a)
class sage.groups.semimonomial_transformations.semimonomial_transformation.SemimonomialTransformation[source]#

Bases: MultiplicativeGroupElement

An element in the semimonomial group over a ring \(R\). See SemimonomialTransformationGroup for the details on the multiplication of two elements.

The init method should never be called directly. Use the call via the parent SemimonomialTransformationGroup. instead.

EXAMPLES:

sage: F.<a> = GF(9)
sage: S = SemimonomialTransformationGroup(F, 4)
sage: g = S(v = [2, a, 1, 2])
sage: h = S(perm = Permutation('(1,2,3,4)'), autom=F.hom([a**3]))
sage: g*h
((2, a, 1, 2); (1,2,3,4), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> 2*a + 1)
sage: h*g
((2*a + 1, 1, 2, 2); (1,2,3,4), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> 2*a + 1)
sage: S(g)
((2, a, 1, 2); (), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> a)
sage: S(1)  # the one element in the group
((1, 1, 1, 1); (), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> a)

>>> from sage.all import *
>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1)
>>> S = SemimonomialTransformationGroup(F, Integer(4))
>>> g = S(v = [Integer(2), a, Integer(1), Integer(2)])
>>> h = S(perm = Permutation('(1,2,3,4)'), autom=F.hom([a**Integer(3)]))
>>> g*h
((2, a, 1, 2); (1,2,3,4), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> 2*a + 1)
>>> h*g
((2*a + 1, 1, 2, 2); (1,2,3,4), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> 2*a + 1)
>>> S(g)
((2, a, 1, 2); (), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> a)
>>> S(Integer(1))  # the one element in the group
((1, 1, 1, 1); (), Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> a)
get_autom()[source]#

Return the component corresponding to \(Aut(R)\) of self.

EXAMPLES:

sage: F.<a> = GF(9)
sage: SemimonomialTransformationGroup(F, 4).an_element().get_autom()
Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> 2*a + 1

>>> from sage.all import *
>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1)
>>> SemimonomialTransformationGroup(F, Integer(4)).an_element().get_autom()
Ring endomorphism of Finite Field in a of size 3^2 Defn: a |--> 2*a + 1
get_perm()[source]#

Return the component corresponding to \(S_n\) of self.

EXAMPLES:

sage: F.<a> = GF(9)
sage: SemimonomialTransformationGroup(F, 4).an_element().get_perm()
[4, 1, 2, 3]

>>> from sage.all import *
>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1)
>>> SemimonomialTransformationGroup(F, Integer(4)).an_element().get_perm()
[4, 1, 2, 3]
get_v()[source]#

Return the component corresponding to \({R^{ imes}}^n\) of self.

EXAMPLES:

sage: F.<a> = GF(9)
sage: SemimonomialTransformationGroup(F, 4).an_element().get_v()
(a, 1, 1, 1)

>>> from sage.all import *
>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1)
>>> SemimonomialTransformationGroup(F, Integer(4)).an_element().get_v()
(a, 1, 1, 1)
get_v_inverse()[source]#

Return the (elementwise) inverse of the component corresponding to \({R^{ imes}}^n\) of self.

EXAMPLES:

sage: F.<a> = GF(9)
sage: SemimonomialTransformationGroup(F, 4).an_element().get_v_inverse()
(a + 2, 1, 1, 1)

>>> from sage.all import *
>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1)
>>> SemimonomialTransformationGroup(F, Integer(4)).an_element().get_v_inverse()
(a + 2, 1, 1, 1)
invert_v()[source]#

Elementwisely invert all entries of self which correspond to the component \({R^{ imes}}^n\).

The other components of self keep unchanged.

EXAMPLES:

sage: F.<a> = GF(9)
sage: x = copy(SemimonomialTransformationGroup(F, 4).an_element())
sage: x.invert_v()
sage: x.get_v() == SemimonomialTransformationGroup(F, 4).an_element().get_v_inverse()
True

>>> from sage.all import *
>>> F = GF(Integer(9), names=('a',)); (a,) = F._first_ngens(1)
>>> x = copy(SemimonomialTransformationGroup(F, Integer(4)).an_element())
>>> x.invert_v()
>>> x.get_v() == SemimonomialTransformationGroup(F, Integer(4)).an_element().get_v_inverse()
True