#15375: Extended Affine Weyl Groups SD40
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Reporter: bump | Owner: bump
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.8
Component: combinatorics | Resolution:
Keywords: days54, coxeter, | Merged in:
days64, days65 | Reviewers: Dan Bump, Anne
Authors: Daniel Bump, Dan | Schilling
Orr, Anne Schilling, Mark | Work issues:
Shimozono, Nicolas Thiery. | Commit:
Report Upstream: N/A | b9152e2bc08cd314744b5fd5ef627a467778f25a
Branch: | Stopgaps:
public/combinat/extended_affine_weyl_groups-15375|
Dependencies: #10963, #14102 |
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Comment (by nthiery):
Replying to [comment:77 mshimo]:
> Nicolas, Thanks very much for your comments.
You are welcome!
> I need to know the reason behind using {{{cartan_type.special_node()}}}.
>
> Let's call this the extra special node. For simplicity we assume
untwisted
> affine type. By deletion we obtain a subsystem of finite type.
> If the extra special node is nonzero then the correct behavior is that
> the "classical subsystem" should be a relabeling of the
> appropriate standard classical subsystem
> with 0 as one of the "finite" Dynkin nodes.
> This nonstandard classical Dynkin node set will get propagated to
> indices of simple roots and fundamental weights and to the
> finite Weyl groups. In particular we could see
> t_{\omega_0^\vee} as a nontrivial translation element
> and s_0 as an element of the finite Weyl group.
> Do we really want this?
It's indeed nice to have 0 by default for the special node in the
canonical labeling of the Dynkin diagram, in order to follow the usual
conventions in the literature.
But other than that the code has been designed to be agnostic
w.r.t. the labels of the nodes of the Dynkin diagram. This gives the
user the flexibility to use his preferred labeling, without imposing a
given convention. Also it makes the code robust in potential use cases
like the following:
- The user inputs some Cartan matrix which turns out to be
affine. There is no reason a priori that 0 will be a special node.
(note: at some point we will need/want to implement the logic to
recover some special node).
- We have a big KM Dynkin diagram, and we want to do some calculations
involving a parabolic subgroup. Typically the corresponding Dynkin
diagram is disconnected, and the calculation will boil down to the
connected pieces. Those connected pieces could be affine Dynkin
diagrams, but with labels like 3,4,5; still we might want to use
it's fundamental group, etc ...
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/15375#comment:78>
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