[sage-combinat-devel] Re: End(CombinatorialFreeModule)?

2018-04-10 Thread Dima Pasechnik


On Thursday, December 6, 2012 at 12:47:53 PM UTC, Dima Pasechnik wrote:
>
> On 2012-12-06, Nicolas M. Thiery  wrote: 
> > On Thu, Dec 06, 2012 at 07:29:57AM +, Dima Pasechnik wrote: 
> >> I wonder if one can actually work in the endomorphism ring/algebra of a 
> >> CombinatorialFreeModule, and if yes, how. 
> > 
> > Not yet. I guess the closest approximation would be to take V \otimes V. 
> > But of course it is not endowed with composition, and it should really 
> > be V^* \otimes V (which makes a difference in non finite dimension). 
> > 
> >> Examples most appreciated. (Ideally, I would like to know how to 
> >> work with algebras specified by multiplication coefficients in this 
> >> framework) 
> > 
> > I am not sure what you mean. Can you be a bit more specific? 
> say, I have a permutation group acting on the basis elements of a 
> CombinatorialFreeModule, and I want to get hold of the endomorphisms 
> commuting with this action. Then it would be natural to represent 
> the ring of such endomorphisms by the multiplication coefficients. 
> (i.e. construct a regular representation of this ring). 
>
>
Another canonical example of "natural" regular representations is the 
quotient of a polynomial ring over a 0-dimensional ideal.
Frankly, it is astonishing that given all the amount of stuff one can do 
with "combinatorial algebras", this is overlooked; this is perhaps the most
basic example of use of linear algebra in computational algebraic geometry, 
after all.

https://groups.google.com/d/msg/sage-support/6Gprakjj1xQ/42E0LYfTBQAJ

 

> Best, 
> Dima 
>
>
> > 
> > Cheers, 
> > Nicolas 
> > -- 
> > Nicolas M. ThiƩry "Isil"  
> > http://Nicolas.Thiery.name/ 
> > 
>
>

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[sage-combinat-devel] Re: End(CombinatorialFreeModule)?

2012-12-06 Thread Andrew Mathas
I think that without some extra information this is bound to be difficult. 
For the modules that I am working with at the moment I can do this and I am 
in the process of making this more explicit.

The modules that I am working with have the following features which make 
this tractable:

   - they are cyclic: G = zA for some z (they are all A-modules)
   - (more importantly) I have a presentation for the modules 
   - the presentation that I have is very efficient in the sense that if 
   I am looking for maps f:G-H then the dimension of the vector subspace of H 
   which can contain f(z) is quite small. This makes it tractable to look at 
   the kernel of the modules relations on this subspace to find a basis for 
   Hom_A(G,H).

Currently, I have code which computes a basis for Hom_A(G,H) and returns 
sage morphisms f:G-H which you can apply to either arbitrary elements of G 
or to the indexing set of the basis for G. Next I plan to have the the code 
automatically write compositions f*g in terms of the distinguished bases 
that I have for these hom-spaces.

As a special case, of course, I can compute End_A(G) an work with this. 
(Although for my examples this is only interesting in characteristic 2.)

Andrew

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