#10527: Implementation of quiver mutation type
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Reporter: stumpc5 | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.3
Component: combinatorics | Resolution:
Keywords: quiver mutation type days38 | Work issues:
Report Upstream: N/A | Reviewers: Hugh Thomas
Authors: Christian Stump | Merged in:
Dependencies: | Stopgaps:
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Comment (by hthomas):
Replying to [comment:114 gmoose05]:
> I just uploaded a new patch which takes care of the issues that Hugh
raised. Thanks for that.
Thanks!
> I also made one or two other changes that will be useful for patches
10538 going forward. Please see my comments below.
It might actually be better to make such changes part of 10538 (I mean, in
the cases that they aren't corrections of actual errors in 10527). This
way, 10527 gets held up while we debate details which it might make more
sense to resolve in the context of where you want to use them. (And
perhaps with a more sympathetic referee:).)
On the whole, I don't think it's a good idea to introduce (A,(m,0), 1) as
a synonym for D,m. It seems to me that introducing synonyms like this is
a process which could go on for ever. There are also wierd corner cases.
I think the only reasonable interpretation of A,(1,0),1 is a loop, and of
A,(2,0),1 is a 2-cycle. So they just *are not* quiver mutation types. It
also seems more confusing than beneficial to me that the quiver which
results is not the quiver which you actually named.
> Yes, you are correct I slightly changed the phrasing for Elliptic, and
then followed your advice for the last three lines. Let me know if this
looks okay.
Yes, it's fine.
> I ensured that AE doesn't take a twist argument, updated the example,
and added a comment to the documentation about the input format. I still
vote to keep it, but don't feel too strongly if you and Christian want to
get rid of it.
I'm not really arguing against it at this point.
> I forget now why we allowed switching the order of k and n. Christian,
do you remember? Anyway, Hugh I agree with you, and made the change.
!QuiverMutationType('GR',(5,2)) for instance now gives an error. But we
still have !QuiverMutationType('GR',(3,5)) =
!QuiverMutationType('GR',(2,5)).
I agree that (2,5) and (3,5) should be the same, certainly!
> >
> > !QuiverMutationType('D',3,2) works, but !QuiverMutationType('A',3,2)
doesn't. But A_3 and D_3 are the same. I think both should yield what
D,3,2 does now (namely, C,2,-1). There has to be some notation for it,
and it's pretty peculiar to require people to think of the root system as
derived from D,3 rather than A,3. (The reason this makes some sense is
that it fits the pattern of D,n,2, not the pattern of A,2n+1,2, but I
think that's okay.)
>
> This was from following Kac's affine notation, which didn't allow
A_3^(2).
Interesting! He really writes D_3^(2)^ and not A_3^(2)^? I'm surprised.
To me, it seems in some sense like a feature not a bug that you can ask
for A_3^(2)^ and you will be confronted with the one thing it has to be,
even if that wasn't what you were expecting.
> However, coercing to ('D',3,2) might be a better choice than an error
message here. Unfortunately, ('A',3,2) doesn't resemble ('A',2k+1,2) in
this case, but perhaps this is okay. Christian, feel free to let us know
if you have a different preference. By the way, I just ran
>
> {{{
> sage: CartanType(['A',3,2])
> ['B', 2, 1]^*
> sage: CartanType(['D',3,2])
> ['C', 2, 1]^*
> sage: CartanType(['A',4,2])
> ['BC', 2, 2]
> sage: CartanType(['A',5,2])
> ['B', 3, 1]^*
> sage: CartanType(['A',7,2])
> ['B', 4, 1]^*
> sage: CartanType(['D',4,2])
> ['C', 3, 1]^*
> sage: CartanType(['D',5,2])
> ['C', 4, 1]^*
> }}}
>
> as a sanity check, and it actually *does differ* from this choice.
In fact !CartanType('B',2,1) and !CartanType('C',2,1) only differ by a
reordering of the roots. This is consistent with our current choice, to
treat !QuiverMutationType('B',2,1) as a synonym for
!QuiverMutationType('C',2,1). It's also consistent with letting
!CartanType('A',3,2) and !CartanType('D',3,2) be the same (since
!QuiverMutationType orders roots differently from !CartanType anyway).
> Should we make
>
> {{{
> sage: QuiverMutationType(['D',3,2])
> ['BB',2,1]
> sage: QuiverMutationType(['A',3,2])
> ['CD',2,1]
> }}}
>
> instead?
No. There is no diagram 'CD',2,1. (In the sense that it isn't drawable.)
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10527#comment:117>
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