[sage-support] Morphisms between free groups.
Dear all, I need to do something which I thought would be quite simple. I have free groups F, G with generators x1,x2,x3,x4 and x,y respectively and I need to compute the image of an element under the homomorphism F-->G:x1->x, x2->y, x3->x, x4->y. I could not find anything in the manual about morphisms between free groups. I tried subs, but that does not work since the parents of the elements of F and G are (of course) different. The workaround I used is to work in the free group H with generators x1,x2,x3,x4,x,y and to use subs. However this is also inconvenient since I need the action of the braid group with 4 strands on F. It does not act on H because of the limitation "#strands=#generators". So this means I have to use only part of the braid group with 6 strands. In the end it all works but it is terribly hacky for some which seems to be a very clean thing to do. I would be grateful for any suggestions. Michel -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-support+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/f7d460d0-390b-455d-ae90-72b105e8e926n%40googlegroups.com.
Re: [sage-support] Morphisms between free groups.
On Fri, Feb 4, 2022 at 4:40 AM 'Michel VAN DEN BERGH' via sage-support
wrote:
>
> Dear all,
>
> I need to do something which I thought would be quite simple. I have free
> groups
> F, G with generators x1,x2,x3,x4 and x,y respectively and I need to compute
> the image of an element under the homomorphism F-->G:x1->x, x2->y, x3->x,
> x4->y.
>
> I could not find anything in the manual about morphisms between free groups.
> I tried subs, but that does not work since the parents of the elements of F
> and G are (of course) different.
>
> The workaround I used is to work in the free group H with generators
> x1,x2,x3,x4,x,y and to use subs. However this is also inconvenient since I
> need the action of the braid group with 4 strands on F. It does not act on H
> because of the limitation "#strands=#generators".
>
> So this means I have to use only part of the braid group with 6 strands.
>
> In the end it all works but it is terribly hacky for some which seems to be a
> very clean thing to do.
>
> I would be grateful for any suggestions.
Michel:
One idea is to check out
https://doc.sagemath.org/html/en/reference/groups/sage/groups/braid.html
Another idea is to download braid-1.1.tar.gz, untar in the pkg
subdirectory and type LoadPackage("braid");
FYI, the pkg subdir is inside sage-9*/local/share/gap
You can find a pdf manual for the braid package at
https://arxiv.org/abs/math/0304376
Currently, the link to braid-1.1.tar.gz at
https://www.gap-system.org/Packages/undep.html
is bad but I have a copy. Just email me privately.
- David
> Michel
>
>
> --
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[sage-support] Re: Eigen spaces of algebraic matrices broken ?
On Apple Mac the example above runs on the 9.4 kernel using either the 9.4
or 9.5 interface but not on the 9.5 kernel from either Interface.
On Thursday, February 3, 2022 at 6:44:47 AM UTC Emmanuel Charpentier wrote:
> Le mercredi 2 février 2022 à 22:15:00 UTC+1, Nils Bruin a écrit :
>
> On Monday, 31 January 2022 at 15:19:49 UTC-8 Emmanuel Charpentier wrote:
>>
>>> As advertised, an atempt at a minimal (non-)working example :
>>>
>>> # Reproducible minimal example
>>> with seed(0): M = matrix(AA, 3, 3, lambda u,v: AA.random_element())
>>> # Working ring
>>> WR = M.base_ring().algebraic_closure()
>>> # A variable to carry the eigenvalues
>>> l = SR.var("l")
>>> # Vector of unknowns for the eigenvectors
>>> V =vector(list(var("v", n=2))+[SR(1)])
>>> # M.eigenvalues does not return. Get them by hand
>>>
>>> Actually, for me on 9.5beta9, `M.eigenvalues()` works just fine.
>>
> Hmmm… You may have obtained a “less pathological” M than I did, due to
> possible differences in random numbers generation (notwithstanding my
> attempt at reproducibility…).
>
> What do you get for M ? I have :
>
> sage: with seed(0): M = matrix(AA, 3, 3, lambda u,v: AA.random_element())
> sage: M.apply_map(lambda u:u.radical_expression())
> [ -sqrt(2) - 1-1/4 -2*sqrt(3)]
> [1/2 1/8*sqrt(33) + 1/8 -1/5*sqrt(29) + 3/5]
> [ 0 1/4 1/2]
>
> So the problem is perhaps just platform-dependent, or there is a very
>> recent change that affected this (my M gets just integer entries from
>> {-2..2})
>>
> Okay. We have a problem in reproducibility : with seed(0): should entail
> a reproducible, platform-independent result. It did not. BTW, what is your
> platform ?
>
> Suggestions on how to document this and file a ticket ?
>
> I agree with the rest of your conclusions, but going to numerical
> approximations then trying to somehow “recognize” the algebraics they are
> approximations of somehow denies the whole point of working in QQbar…
>
> Looking at the example a bit: you'd be forcing sage to work with a huge
>> compositum if you're actually getting a 3x3 matrix with non-rational
>> algebraic entries: even if they are just independent quadratics, you'd end
>> up in an extension of degree 2^9. This will only work in very limited cases.
>>
>> One way to get this kind of thing to work is to work with high-precision
>> floats, use numerically (fairly) stable methods to compute the desired
>> answer, and then try to recognize it as algebraic. You probably only care
>> if it is one of fairly low height. You can then try to turn your
>> computation into proof, possibly by tracing through height bounds and
>> showing your precision was sufficient to identify the right solution
>> uniquely.
>>
>> You could work on automating this kind of thing, but I doubt you'd ever
>> get it to work on a reasonable range of examples; just because the height
>> bounds would be rather ill-behaved.
>>
>> You can still trace the root cause further on this and perhaps improve
>> arithmetic in AA a bit, but the general shape of the problem you're trying
>> to deal with does not look promising for generally performant methods.
>>
>
>
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Re: [sage-support] Morphisms between free groups.
On Fri, 4 Feb 2022, 11:02 David Joyner, wrote:
> On Fri, Feb 4, 2022 at 4:40 AM 'Michel VAN DEN BERGH' via sage-support
> wrote:
> >
> > Dear all,
> >
> > I need to do something which I thought would be quite simple. I have
> free groups
> > F, G with generators x1,x2,x3,x4 and x,y respectively and I need to
> compute the image of an element under the homomorphism F-->G:x1->x, x2->y,
> x3->x, x4->y.
> >
> > I could not find anything in the manual about morphisms between free
> groups. I tried subs, but that does not work since the parents of the
> elements of F and G are (of course) different.
> >
> > The workaround I used is to work in the free group H with generators
> x1,x2,x3,x4,x,y and to use subs. However this is also inconvenient since I
> need the action of the braid group with 4 strands on F. It does not act on
> H because of the limitation "#strands=#generators".
> >
> > So this means I have to use only part of the braid group with 6 strands.
> >
> > In the end it all works but it is terribly hacky for some which seems to
> be a very clean thing to do.
> >
> > I would be grateful for any suggestions.
>
> Michel:
>
> One idea is to check out
> https://doc.sagemath.org/html/en/reference/groups/sage/groups/braid.html
>
> Another idea is to download braid-1.1.tar.gz, untar in the pkg
> subdirectory and type LoadPackage("braid");
>
most, if not all, functionality of braid is nowadays in
https://gap-packages.github.io/MapClass/
(which is in our gap_packages)
FYI, the pkg subdir is inside sage-9*/local/share/gap
> You can find a pdf manual for the braid package at
> https://arxiv.org/abs/math/0304376
> Currently, the link to braid-1.1.tar.gz at
> https://www.gap-system.org/Packages/undep.html
> is bad but I have a copy. Just email me privately.
>
> - David
>
> > Michel
> >
> >
> > --
> > You received this message because you are subscribed to the Google
> Groups "sage-support" group.
> > To unsubscribe from this group and stop receiving emails from it, send
> an email to sage-support+unsubscr...@googlegroups.com.
> > To view this discussion on the web visit
> https://groups.google.com/d/msgid/sage-support/f7d460d0-390b-455d-ae90-72b105e8e926n%40googlegroups.com
> .
>
> --
> You received this message because you are subscribed to the Google Groups
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> .
>
--
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