#13726: The semimonomial group
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Reporter: tfeulner |
Owner: joyner
Type: enhancement |
Status: new
Priority: major |
Milestone: sage-5.6
Component: group theory |
Resolution:
Keywords: (semi-)monomial group, semilinear action, isometry group |
Work issues:
Report Upstream: N/A |
Reviewers:
Authors: Thomas Feulner |
Merged in:
Dependencies: |
Stopgaps:
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Description changed by tfeulner:
Old description:
> A semimonomial group over a ring `R` of length `n` is equal to the wreath
> 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^{-1}(\psi_{\pi(0} ) ), \ldots,
> \alpha^{-1}(\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.
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^{-1}(\psi_{\pi(0} ) ), \ldots,
\alpha^{-1}(\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.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/13726#comment:1>
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