#14085: Ambient spaces for dual and affine root systems
-------------------------------------+--------------------------------------
Reporter: nthiery | Owner: sage-combinat
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.8
Component: combinatorics | Resolution:
Keywords: | Work issues:
Report Upstream: N/A | Reviewers: Dan Orr, ...
Authors: Nicolas M. Thiéry | Merged in:
Dependencies: | Stopgaps:
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Description changed by nthiery:
Old description:
> This ticket implements ambient spaces for dual and affine root
> systems. So now all finite and affine crystallographic root systems
> have an ambient space, except for relabeled ones (to be handled in a
> separate ticket).
>
> Upcoming applications:
>
> * Plotting #4237
> * Affine permutation representations built from the ambient space
> * Macdonald polynomials
> * Affine weight ring (in waiting for #7922 for weight rings built from
> the root or weight lattice)
>
> A couple things handled by this patch:
> - Fixing a couple whitespaces
> - Standard color scheme (to be used by #4237)
> - Fundamental weights computed from simple roots (finite type)
> - Better handling of the smallest base ring for the ambient space
> (toward ambient spaces over fields/rings like RR)
> - Better handling of dual types: in particular make sure the dual
> CartanType belongs to the appropriate CartanType classes; this
> allowed to remove now unnecessary code.
> - Fix the embedding of the root lattice into the extended weight
> lattice for type BC dual. A Type free approach would be better.
> - Prevent affine crystals to use the ambient space, even if available,
> for backward compatibility. This may be changed in a later ticket.
New description:
This ticket implements ambient spaces for dual and affine root
systems. So now all finite and affine crystallographic root systems
have an ambient space, except for relabeled ones (to be handled in a
separate ticket).
Upcoming applications:
* Plotting #4237
* Affine permutation representations built from the ambient space
* Macdonald polynomials
* Affine weight ring (in waiting for #7922 for weight rings built from
the root or weight lattice)
A couple things handled by this patch:
- Fixing a couple whitespaces
- Standard color scheme (to be used by #4237)
- Fundamental weights computed from simple roots (finite type)
- Better handling of the smallest base ring for the ambient space
(toward ambient spaces over fields/rings like RR)
- Better handling of dual types: in particular make sure the dual
CartanType belongs to the appropriate CartanType classes; this
allowed to remove now unnecessary code.
- Fix the embedding of the root lattice into the extended weight
lattice for type BC dual. A Type free approach would be better.
- Prevent affine crystals to use the ambient space, even if available,
for backward compatibility. This may be changed in a later ticket.
- Implement to_simple_root, and generalizes associated_reflection for
every root lattice realizations using it
- More tests for the null (co)root.
- Improved section about ambient spaces in the introduction to root
systems.
Includes the link with (affine/signed) permutation representations.
- Improved doc for the delta notation glitch for extended weight lattice
realizations.
- Various documentation improvements.
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14085#comment:7>
Sage <http://www.sagemath.org>
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