[sage-combinat-devel] Re: End(CombinatorialFreeModule)?
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. Thiéry "Isil" > > http://Nicolas.Thiery.name/ > > > > -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-combinat-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-combinat-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-combinat-devel. For more options, visit https://groups.google.com/d/optout.
[sage-combinat-devel] Re: End(CombinatorialFreeModule)?
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). Best, Dima > > Cheers, > Nicolas > -- > Nicolas M. Thiéry "Isil" > http://Nicolas.Thiery.name/ > -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To post to this group, send email to sage-combinat-devel@googlegroups.com. To unsubscribe from this group, send email to sage-combinat-devel+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sage-combinat-devel?hl=en.
[sage-combinat-devel] Re: End(CombinatorialFreeModule)?
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 -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To view this discussion on the web visit https://groups.google.com/d/msg/sage-combinat-devel/-/HeFrlWzYFXQJ. To post to this group, send email to sage-combinat-devel@googlegroups.com. To unsubscribe from this group, send email to sage-combinat-devel+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sage-combinat-devel?hl=en.