#13726: The semimonomial group
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Reporter: tfeulner | Owner: joyner
Type: enhancement | Status:
Priority: major | positive_review
Component: group theory | Milestone: sage-5.13
Keywords: (semi-)monomial group, | Resolution:
semilinear action, isometry group | Merged in:
Authors: Thomas Feulner | Reviewers: Volker
Report Upstream: N/A | Braun
Branch: | Work issues:
Dependencies: | Commit:
| Stopgaps:
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Changes (by vbraun):
* status: needs_review => positive_review
* reviewer: => Volker Braun
Old description:
> A semimonomial group over a ring `R` of length `n` is equal to the
> semidirect product of the monomial group and the group of ring
> automorphisms. The multiplication of two elements `(\phi, \pi,
> \alpha)(\psi, \sigma, \beta)` with
>
> * `\phi, \psi \in {R^*}^n`
> * `\pi, \sigma \in S_n`
> * `\alpha, \beta \in Aut(R)`
>
> is defined by:
>
> `(\phi, \pi, \alpha)(\psi, \sigma, \beta) := (\phi * \psi^{\pi,
> \alpha}, \pi * \sigma, \alpha * \beta)`
>
> with
>
> `\psi^{\pi, \alpha} := (\alpha(\psi_{\pi(0} ) ), \ldots,
> \alpha(\psi_{\pi(n-1} ) ) )`
>
> and an elementwisely defined multiplication of vectors.
>
> This group plays an important role in coding theory since it is the group
> of all semilinear isometries (relative to the Hamming/Lee/homogenous
> metric) of the ambient space.
>
> ----
> apply only: trac_13726-semimonomial_group.2.patch
New description:
A semimonomial group over a ring `R` of length `n` is equal to the
semidirect product of the monomial group and the group of ring
automorphisms. The multiplication of two elements `(\phi, \pi,
\alpha)(\psi, \sigma, \beta)` with
* `\phi, \psi \in {R^*}^n`
* `\pi, \sigma \in S_n`
* `\alpha, \beta \in Aut(R)`
is defined by:
`(\phi, \pi, \alpha)(\psi, \sigma, \beta) := (\phi * \psi^{\pi, \alpha},
\pi * \sigma, \alpha * \beta)`
with
`\psi^{\pi, \alpha} := (\alpha(\psi_{\pi(0} ) ), \ldots,
\alpha(\psi_{\pi(n-1} ) ) )`
and an elementwisely defined multiplication of vectors.
This group plays an important role in coding theory since it is the group
of all semilinear isometries (relative to the Hamming/Lee/homogenous
metric) of the ambient space.
----
apply only: trac_13726-semimonomial_group_vb.patch
--
--
Ticket URL: <http://trac.sagemath.org/ticket/13726#comment:17>
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