implement the cluster fans

[chapoton]

Fans play an important role in cluster algebra theory. Let's have them in.

[chapoton]

New commits:

​66173bd

trac #17947 implement the cluster fans

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​07cd33c

trac #17947 details about cluster fan

[darij]

Care to define it in the docstring?

I've always wanted to understand what Nathan is talking about all the time, but I never got over my laziness to read up on it. I guess this is my chance :) Thanks a lot.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9628ba3

trac #17947 more doc

[darij]

Thanks -- this makes things a lot less scarier. I feel I need some more help, though. Is a simplicial fan the same as a simplicial complex but with "simplex" replaced by "simplicial cone"? How exactly do the vectors d in \ZZ^n give rise to a cone? Is the cone corresponding to a cluster simply the set of all nonnegative linear combinations of the denominator vectors? What is a smooth fan? Why do you require acyclicity; isn't finite-type enough?

[chapoton]

Replying to darij:

Thanks -- this makes things a lot less scarier. I feel I need some more help, though.

Yes. I hope you do not want all these explanations to be written in the doc ?

Is a simplicial fan the same as a simplicial complex but with "simplex" replaced by "simplicial cone"?

Yes, essentially, but not quite because some convexity issues enter the definition. For the exact definition of a fan, see Fulton book on toric varieties for example. A simplicial fan is a fan where every cone is simplicial, namely has the minimal number of generators wrt its dimension.

How exactly do the vectors d in \ZZ^n give rise to a cone?

The d associated with variables in a given cluster span the rays of a cone.

Is the cone corresponding to a cluster simply the set of all nonnegative linear combinations of the denominator vectors?

Yes.

What is a smooth fan?

a smooth fan is a simplicial fan such that each cone is smooth. A smooth cone is a simplicial cone whose rays define a basis of the lattice (they have determinant +-1)

This is related to the smoothness of the toric variety associated to the fan.

Why do you require acyclicity; isn't finite-type enough?

No, with this version of the fan. If you apply it to the cyclic quiver with 3 vertices, the result is not an interesting fan (in particular it is not smooth). There is maybe another version of the fan using the g-vector instead. I do not remember the details right now.

[darij]

OK, I am not able to review this, but thank you a lot for the explanations. I still don't know what almost positive roots are, how Cone infers its lattice, etc., so I'll defer to someone more familiar with these things, though the code does look good to me.

---

New commits:

​db09838	Merge branch 'u/chapoton/17947' of git://trac.sagemath.org/sage into fan
​fb98b4d	doc improvements to quiver.py:

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​82ed222

trivial doc fix

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c3b9534	Merge branch 'develop' into t/17947/public/ticket/17947
​b6e225a	Add d-vector to ClusterSeed
​d18903f	Add d-matrix to ClusterSeed
​b2a8278	Removed trailing spaces
​9f56277	Merge remote-tracking branch 'origin/u/etn40ff/add_d_vectors' into t/17947/public/ticket/17947
​df182e5	I changed the name of the function fan to d_vector_fan.

[vpilaud]

I attempt my first Sage Days on cluster algebras, and I wanted to implant d-vector fans and g-vector fans. I realized that this was already partially done in this ticket. For the functions d_vector_fan and g_vector_fan to be coherent, I merged with #18588 which implements the d_matrix function.

[jipilab]

Hi!

I would be happy to review this ticket!

It will be quite handy!

[chapoton]

Duplicate import

+        from cluster_seed import ClusterSeed
+        from cluster_seed import ClusterSeed

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​0eea07b

remove duplicate import, as well as a non-necessary import

[jipilab]

There are blank lines that contain white spaces in the function d_vector_fan in the quiver.py file.

Otherwise it looks good to me.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​9b2eb28	Merge branch 'public/ticket/17947' into 6.8.b6
​2b1e105	trac #17947 taking care of trailing whitespaces

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​74a57fc

Merge branch 'public/ticket/17947' into 6.9.b6

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​97214cf

Merge branch 'public/ticket/17947' into 6.9.rc0

[chapoton]

Ping ? The patchbot has given a green light !

[chapoton]

Re-ping ? this should be an easy ticket !

[jipilab]

All test passed on sage-6.9.

This looks good to me. I set this as positive review.

[chapoton]

Thanks a lot ! Maybe you could also look at #16209 ?

[vbraun]

Just a PSA: Your ticket will never be merged if it depends on a invalid/wontfix ticket. At least not until that one is being merged into the git tree.

[chapoton]

ok, thanks. Let us remove the dependency then. It has been taken care of in another ticket.