#17947: implement the cluster fans
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Reporter: | Owner:
chapoton | Status: needs_review
Type: | Milestone: sage-6.6
enhancement | Resolution:
Priority: minor | Merged in:
Component: | Reviewers:
combinatorics | Work issues:
Keywords: | Commit:
cluster, fan | 9628ba3adff463a4d57a030bd75a57ae2b7aea44
Authors: | Stopgaps:
Frédéric Chapoton |
Report Upstream: N/A |
Branch: |
u/chapoton/17947 |
Dependencies: |
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Comment (by chapoton):
Replying to [comment:5 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.
--
Ticket URL: <http://trac.sagemath.org/ticket/17947#comment:6>
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