#10527: Implementation of quiver mutation type
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Reporter: stumpc5 | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.1
Component: combinatorics | Resolution:
Keywords: quiver mutation type days38 | Work issues:
Report Upstream: N/A | Reviewers: Hugh Thomas
Authors: Christian Stump | Merged in:
Dependencies: #10349 | Stopgaps:
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Comment (by hthomas):
Hi Nicolas and Christian--
The only relevant reference I found to Bourbaki is in
root_system/cartan_type.py where it says that the *numbering* of roots
follows Bourbaki. What I am saying is not contrary to this.
In sage-combinat-devel, in June of 2008, in the thread "tensor product of
crystal code", I found Dan Bump saying the following:
> It says so in the documentation:
> > The edge multiplicities are encoded as edge labels. This uses the
> > convention in Kac / Fulton Harris Representation theory
> > wikipedia http://en.wikipedia.org/wiki/Dynkin_diagram, that is for i
!= j:
> >
> > j -k-> i <==> a_ij = -k <==> -scalar(coroot[i],root[j]) = k
> > <==> multiple arrows point from the longer root to the shorter one
>
> That is the standard convention but in fact what is implemented is
>
> j -k-> i <==> a_ji = -k
>
> See this line in cartan_matrix.py():
>
> cmf = lambda i,j: 2 if i == j else -f(j,i)
Since Kac and Bourbaki are contradictory (according to F&Z), everything is
consistent.
Dan seems to be unaware that the Kac convention which he calls "standard"
is contrary to Bourbaki, but this would be an easy thing not to be aware
of.
I still think it would be useful to say explicitly in the cartan_matrix
methods of !CartanType and !QuiverMutationType that the cartan_matrix is
according to !Kac/Fulton-Harris and contrary to Bourbaki.
I did also double-check the Dynkin diagrams, and they are correct
consistent with this.
And just to clarify, that means that all the finite type
!QuiverMutationTypes are consistent with what was already in Sage and
consistent with Fomin and Zelevinsky (which is not surprising -- otherwise
I'm sure it would have been picked up on already).
cheers,
Hugh
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10527#comment:65>
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