#14014: Update matrix groups to new Parents, libGAP.
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Reporter: vbraun | Owner: joyner
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.9
Component: group theory | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers: David Roe
Authors: Volker Braun | Merged in:
Dependencies: #14187, #14323 | Stopgaps:
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Comment (by dimpase):
Replying to [comment:18 vbraun]:
> === comment:15 ===
>
> I just want to allow the GMP backend for long integers, I don't think
> we gain anything from trying to support the old home-grown one.
I agree, I'm just asking for a bit more explanation there.
> Though
> at this point you can still use both, the conversion from long
> integers to Sage is done via strings. But I'd like to change that in
> the future and directly access the GMP limbs.
>
> The libgap before this ticket doesn't handle long its correctly
> because I didn't understand what GAP docs mean by "immediate
> integers". This is why I wrote the explanatory comment.
>
> === comment:16 ===
>
> That file has been renamed to just `morphism.py`, if you have all
> patches you might have to run `sage -sync-build`
Hmm, I tried, and it didn't help. I'll build a pristine installation of
Sage and test this there.
>
> === comment:17 ===
>
> I agree that there is no unique bilinear scalar form, though I would
> call **the** orthogonal group (in absence of any further qualifiers)
> over a ring the one with the standard (identity matrix) bilinear
> form.
Actually, it's not even correct to talk about bilinear forms; one should
talk about quadratic forms,
in case you do not want to exclude the case of fields of characteristic 2
(it's more or less a coincidence that
in other characteristics the automorphisms of the associate bilinear form
coincide with the (linear) automorphisms of the quadratic form.
As well, in characteristic 2 orthogonal matrices have little to do with
orthogonal groups; indeed they fix the quadratic form `f(X)=<X,X>`, but
this form is (sesqu)linear, i.e. `f(aX+bY)=<aX+bY,aX+bY>=a^2
<X,X>+b^2<Y,Y>+2<aX,bY>= a^2<X,X>+b^2<Y,Y>`, and this is not a one one
cares much about. The ones which are interesting cannot even be
diagonalized.
And what happens for rings in general is much more mysterious, even for
ZZ. I don't know much about this topic, but number theorists might start
pulling their hairs out upon reading this part; already binary quadratic
forms over ZZ are a big topic.
> In particular since the orthogonal groups for other bilinear
> forms have sometimes other names, as you said. The next paragraph
> explains how the notation for finite fields differ, so I think thats
> clear enough:
> {{{
> The general orthogonal group `GO(n,R)` consists of all orthogonal
> `n\times n` matrices over the ring `R`.
>
> In the case of a finite field and if the degree `n` is even, then
> there are two inequivalent quadratic forms and a third parameter
> ``e`` must be specified to disambiguate these two possibilities.
> }}}
> My aim was to start the docstring with a friendly
> paragraph that says what GO/SO is before hitting the user with the
finite
> field technicalities.
Well, one has to start with a correct paragraph.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14014#comment:19>
Sage <http://www.sagemath.org>
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