#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
---------------------------+------------------------------------------------
Changes (by stumpc5):
* cc: stumpc5, saliola (added)
* dependencies: 12412, 12413 => #12412, #12413
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.
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.
Let me know what you think,
best, Christian
--
Comment:
I just had a quick look at the patch, it seems to be very complete -
thanks!
We are currently slowly getting a large project into sage dealing with
(the combinatorics of) cluster algebras and quivers, see #10298. In
particular, we also have a class quiver (#10538). I think both classes
should be merged as they both deal with acyclic quivers. To be more
precice, we deal with skew-symmetrizable matrices and the corresponding
labeled quivers, while you seem to focus on the simply-laced version.
I can definitely to the review for this patch, and I propose to sit down
(virtually together) and merge both concepts. This means make one being
dependent of the other.
I also want to remark that Franco Saliola has as well quite some code on
quiver representations that we might want to merge (if it is not
subsumed). We also have currently a project of implementing the knitting
algorithm for quivers without cycles which is currently based on Franco's
implementation.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12630#comment:2>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
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