#12630: Add representations of quivers and quiver algebras to sage
---------------------------+------------------------------------------------
Reporter: JStarx | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.0
Component: algebra | Keywords: algebra, quiver, module
Work_issues: | Upstream: N/A
Reviewer: | Author: JStarx
Merged: | Dependencies: #12412, #12413
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Old description:
> This will add classes dealing with quivers, quiver algebras,
> representations of quivers, elements of these representations,
> homomorphisms between these representations, and spaces of homomorphisms
> between these representations.
>
> There's a lot here that is really easily computable. We can compute
> socles, quotients, radicals, duals, and more for any finite dimensional
> representation. We can compute projective covers of modules so
> Auslander-Rieten translations have been implemented and there's certainly
> potential for future enhancements dealing with homology and cohomology.
> There's only so much I can say here but everything is fully documented
> and should be self explanatory.
>
> Two shortcomings are that quivers need to be acyclic (to keep things
> finite dimensional) and this code does not handle quivers with relations.
> As far as quivers with relations go there are comments in the code
> detailing what should be done to implement that. It's well within the
> reach of Sage, I just don't have the time to do it at the moment.
>
> Let me know what you think,
>
> best, Christian
New description:
This will add classes dealing with quivers, quiver algebras,
representations of quivers, elements of these representations,
homomorphisms between these representations, and spaces of homomorphisms
between these representations.
There's a lot here that is really easily computable. We can compute
socles, quotients, radicals, duals, and more for any finite dimensional
representation. We can compute projective covers of modules so Auslander-
Rieten translations have been implemented and there's certainly potential
for future enhancements dealing with homology and cohomology. There's
only so much I can say here but everything is fully documented and should
be self explanatory.
Two shortcomings are that quivers need to be acyclic (to keep things
finite dimensional) and this code does not handle quivers with relations.
As far as quivers with relations go there are comments in the code
detailing what should be done to implement that. It's well within the
reach of Sage, I just don't have the time to do it at the moment.
--
Comment(by JStarx):
Correct me if I'm wrong, but my understanding is that a simply-laced
quiver is a quiver whose underlying undirected graph is Dynkin type A, D,
or E. If this is what you meant then no, this patch doesn't focus on
simply-laced quivers. Any finite acyclic quiver is allowed, it could have
multiple edges, be disconnected, it doesn't need to be Dynkin or even
affine Dynkin.
Also it's important that the quivers in my patch have unique
representation in Sage because they are part of the defining data of a
parent, whereas the point of the combinat quivers is that they can be
mutated. So I'm not sure combining the two classes makes sense. I would
be very interested in what Simon has to say about this (and in general
about my patch) from an algebra/representation theory perspective, as I'm
pretty new to Sage development.
Franco: I hadn't really considered the possibility till now. But with
funding I would definitely be able to come. I'll go ahead and apply.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12630#comment:5>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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