unstable modules + creating a small library

[Pierre]

hi all

I've just realized that SAGE knows about the Steenrod algebra now.
Does it know about unstable modules, too ?

I have another, related question. I have computed the unstable module
structure on the mod 2 cohomology rings of quite a bunch of finite
groups, see

http://www-irma.u-strasbg.fr/~guillot/research/cohomology_of_groups/index.html

I was thinking that I should, somehow, provide a file readable by SAGE
so that people could use these algebras. For one thing it would
provide many examples of unstable modules, which is always good to
test ideas about the Steenrod algebra. And regardless of the steenrod
operations, even the cohomology rings, as computed by Carlson and
others, are not available in SAGE yet (they're there as Magma files).
At this point I can relatively easily provide a partial translation
into SAGE.

However I was wondering about the best "format" for this: assuming the
unstable algebra class does not exist, shall I present the algebras as
quotients of polynomial rings ? or just give a couple of SAGE lists
with the generators and relations, possibly just members of the formal
ring ? or something pickled perhaps ? I really don't know. Note that
I've got more information on these algebras yet (Stiefel-Whitney
classes...)

And shall I think of a mechanism for people to download ALL the
examples at once rather than separately ? (perhaps useful to try a
conjecture about unstable modules ?)

suggestions most welcome.
Thanks,

pierre

[John H Palmieri]

On Jan 19, 6:28 am, Pierre <pierre.guil...@gmail.com> wrote:
> hi all
>
> I've just realized that SAGE knows about the Steenrod algebra now.
> Does it know about unstable modules, too ?

No, it doesn't, unfortunately. (Sage doesn't know about tensor
products, which has delayed me from implementing various things, like
the coproduct, and I suppose modules.)

> I have another, related question. I have computed the unstable module
> structure on the mod 2 cohomology rings of quite a bunch of finite
> groups, see
>

> http://www-irma.u-strasbg.fr/~guillot/research/cohomology_of_groups/i...

This looks very nice.

> I was thinking that I should, somehow, provide a file readable by SAGE
> so that people could use these algebras.

Sure.

> For one thing it would
> provide many examples of unstable modules, which is always good to
> test ideas about the Steenrod algebra. And regardless of the steenrod
> operations, even the cohomology rings, as computed by Carlson and
> others, are not available in SAGE yet (they're there as Magma files).
> At this point I can relatively easily provide a partial translation
> into SAGE.

I think Simon King does some group cohomology computations with Sage,
but I don't know exactly how he does it.

> However I was wondering about the best "format" for this: assuming the
> unstable algebra class does not exist, shall I present the algebras as
> quotients of polynomial rings ? or just give a couple of SAGE lists
> with the generators and relations, possibly just members of the formal
> ring ? or something pickled perhaps ? I really don't know. Note that
> I've got more information on these algebras yet (Stiefel-Whitney
> classes...)

It sounds to me as though you should create a new class, the
UnstableAlgebra class, or the ModularGroupCohomology class, or
something, which should derive from the class of quotients of
polynomial algebras (so you can define at least part of the structure
by specifying such a quotient), and then there should be extra
structure: the Steenrod operations and Stiefel-Whitney classes and
whatever else you have.

> And shall I think of a mechanism for people to download ALL the
> examples at once rather than separately ? (perhaps useful to try a
> conjecture about unstable modules ?)

You might have two files: one which defines the class, and another
which presents all of the examples. I haven't used databases in Sage,
but perhaps the examples could be a dictionary indexed by the group,
or something like that?

I'm looking forward to whatever you come up with.

John Palmieri

[Pierre]

thanks for your advice. I'll try to write a new class, but i won't try
to merge it into SAGE ! i'll just try to use reasonable names for the
methods, so that my examples remain compatible with whatever
improvement gets written for SAGE one day... for the "database" i'll
use a dict indexed by the "group ID" as GAP understands it.

will keep you posted, but don't hold your breath, i'm afraid i have a
lot to do these days !

pierre

[Simon King]

Dear Pierre, dear John,

John H Palmieri schrieb:

> On Jan 19, 6:28�am, Pierre <pierre.guil...@gmail.com> wrote:
> I think Simon King does some group cohomology computations with Sage,
> but I don't know exactly how he does it.

Indeed. One of my plans is to enrich my results by Steenrod actions,
and to provide the cohomology rings in a Sage readable data base.

My approach for computing cohomology rings of p-groups is:
- use David Green's approach to compute minimal projective resolutions
- I wrote various Cython modules for Sage that compute the visible
ring structure degree by degree
- Use an improved version of Dave Benson's completeness criterion.

Our results:
- We can compute the cohomology of all groups of order 64 in a total
of less than 30 CPU-minutes.
- We computed the cohomology for all groups of order 128
- We computed all but 7 cohomology rings for 3-, 5- and 7-groups up to
order 625
- We computed the cohomology for the Sylow-2-subgroup of the Higman-
Sims group (verifying Carlson's computation)
- We were the first to compute the cohomology of the Sylow-2-subgroup
of the third Conway group

Our results are available at
http://users.minet.uni-jena.de/~king/cohomology/

We plan to make a Sage package out of our programs.

And, Pierre, by the way, I did my PhD in Strasbourg at IRMA (my
advisor was Vladimir Turaev).

Cheers,
Simon