#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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Comment(by stumpc5):

 Replying to [comment:5 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.

 Starting with any quiver, you can construct a skew-symmetric matrix
 (m_{i,j}) with m_{i,j} is the number of edges from i to j minus the number
 of edges from j to i (one is always supposed to be 0). That's what I meant
 with simply-laced. One can also think of quivers (like in type B) coming
 from a skew-symmetrizable matrix (i.e., M such that DM is skew-symmetric
 for a positive diagonal matrix D), where the edges are labeled
 (m_{i,j},m_{j,i}). Our "combinatorial quiver" can be construced for any
 skew-symmetrizable matrix (not in any way restriced to finite or affine
 types).

 > 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.

 This might be right. We could also think of two classes "Quiver" and
 "CombinatorialQuiver" (as your construction is really the one Gabriel
 named quiver), and then having maps between them. Can you think of any
 interaction between the two concepts (like mutating in sinks and sources
 corresponds in finite types to picking a particular Coxeter element, which
 corresponds to picking a particular Auslander-Reiten quiver -- we are also
 currently preparing a paper where we give an alternative combinatorial
 view on Auslander-Reiten translates in finite types -- any further ideas,
 maybe beyond finite types)?

 Best, Christian

-- 
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12630#comment:6>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica, 
and MATLAB

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