#12630: Add representations of quivers and quiver algebras to sage
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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>
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