[GAP Forum] Question about obtaining the PIMs of a finite group algebra kG in a quick way and MeatAxe64

[Jürgen Müller]

Dear Bernhard,

> Rumors told that there is MeatAxe64, but I don't have many pieces of
> information about this yet. Is it freely available?

About this, you probably best ask Richard Parker himself.

> I don't want to chop the regular module, and I am speaking of groups of
> order approximately between 20 000 and 1000 000.

Looking for PIMs is a matter of spinning up vectors (in MeatAxe language)
in suitable modules, rather than chopping. But PIMs typically are `large’ 
modules, whose dimension is comparable to the group order, so that you 
might have to deal with modules `close’ to the regular module.

Of course, spinning in the regular module in principle yields a general 
algorithm, which even might be feasible in your cases, since the regular 
module is a permutation module.

But, depending on the example under consideration, you might be able 
to do better: for example, if G is p-solvable, then the 1-PIM is just the 
permutation module on the cosets of Hall p’-subgroup.

> Later, I would like to use the obtained result tomake further computations
> with other kG-modules, as well.

Doing sophisticated computations with modules of dimension 10^6
might easily be prohibitive.  So, what precisely do you want to know 
about the PIMs? (For example, if you are interested in Ext groups 
between simple modules, there are other ways to tackle that.)

Hope this helps.

Best wishes, Jürgen Müller

> Am 21.10.2020 um 00:09 schrieb Bernhard Boehmler <bernhard.boehmler at googlemail.com>:
> 
> Dear GAP forum,
> 
> let G be a finite group and k be a finite field where char(k)=p divides the
> order of G.
> 
> I would like to kindly ask you the following question:
> 
> Is there an algorithm (freely available or already implemented in GAP4)
> that can find in a relatively quick way the PIMs of kG for relatively big
> groups G?
> 
> I wouldn't care if it took a few days, but a few months would probably be
> too time consuming.
> 
> I don't want to chop the regular module, and I am speaking of groups of
> order approximately between 20 000 and 1000 000.
> 
> Later, I would like to use the obtained result tomake further computations
> with other kG-modules, as well.
> 
> Rumors told that there is MeatAxe64, but I don't have many pieces of
> information about this yet. Is it freely available?
> 
> Any help is appreciated.
> 
> Thanks in advance.
> 
> Kind regards,
> Bernhard Boehmler
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