#12630: Add representations of quivers and quiver algebras to sage
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Reporter: JStarx | Owner:
Type: enhancement | AlexGhitza
Priority: major | Status:
Component: algebra | needs_work
Keywords: algebra, quiver, module, | Milestone: sage-5.13
days49 | Resolution:
Authors: Jim Stark, Simon King, | Merged in:
Mathieu Guay-Paquet, Aladin Virmaux | Reviewers: Simon
Report Upstream: N/A | King
Branch: | Work issues:
Dependencies: #12412, #12413, #14806 | Commit:
| Stopgaps:
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Comment (by nthiery):
Hi Simon,
We just discussed with Nathann; for whatever it's worth, here is my
feeling for the user interface aspects:
Replying to [comment:132 SimonKing]:
> I somehow agree. An argument against "path magma" was the fact that this
particular magma is associative. And "associative path magma" sounds not
good to me.
>
> > But well.. It's not mine to decide `^^;`
> >
> > What about `path_monoid` (if it is a monoid) ? `:-P`
>
> This would have been my choice, too. But sadly it is a monoid if and
only if the quiver has precisely one vertex (which is certainly not always
the case).
At this point I would go for either path_partial_monoid (if it
includes the empty path as identity) or path_partial_semigroup. It
will be easily found by someone interested by paths, and the "monoid"
or "semigroup" piece will highlight the associative multiplicative
structure and the presence, or not, of an identity.
I am even tempted by the shorter path_monoid or path_semigroup.
Granted, this is an abuse. But this abuse is pretty obvious to anyone
familiar with paths endowed with the concatenation product; also I
have often heard this abuse made by mathematicians. As for those that
don't know that are not familiar with path concatenation we can hope
that they will read the documentation where this will be written in
bold.
Just decide among those four the one that pleases you most!
> > > - `.quiver_algebra(R)` (or just `.algebra(R)`?): Return the monomial
algebra
> > > over R whose monomials are given by the free small category
> >
> > What about `path_alebra` ?
> > http://en.wikipedia.org/wiki/Quiver_(mathematics)#Path_algebra
>
> I'd totally agree. However, note that the original code is due to Jim
Stark, and he chose to call it quiver algebra/representation. I refactored
the code and added "free small category", but I think it would be undue to
change the terminology of the pre-existing stuff.
I like path_algebra too, and it's consistent with the above. I'd say
it's fair at this point to change this name.
> > > - `.quiver_representation(...)`: Create a module over the quiver
algebra.
> >
> > To me this should be `.quiver_algebra().representation()`.
>
> To me as well. But it seems that some people really think of it in terms
of the graph.
My feeling:
- Provide the feature as ....algebra.representation() now.
- Add .quiver_representation() later, as an alias, if there really is a
popular request for it.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/12630#comment:133>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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