#16087: Limit roots in the geometric representation of Coxeter groups
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Reporter: nthiery | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.2
Component: group theory | Resolution:
Keywords: days57, plot, coxeter | Merged in:
Authors: Jean-Philippe Labbé, Vivien | Reviewers:
Ripoll | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: #15703 |
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Changes (by jipilab):
* dependencies: => #15703
Old description:
> Related tickets:
> - #9290
> - #13248
> - #15974
>
> TODO:
> - Cleanup this description!!!!!
>
> Matrix representation for Coxeter groups
> - [ ] Root space for general Coxeter groups
>
> sage: L = RootSystem(<"generalized" coxeter matrix>).root_space()
> sage: W = L.reflection_group() returns an instance of
> "CoxeterMatrixGroup"
>
> - [ ] Generalize CoxeterMatrixGroup from WeylGroup
> http://trac.sagemath.org/ticket/15703
> Make it full featured
> - [ ] Add a category for Lorenzian/HyperbolicCoxeterGroups
> Inside the hierarchy:
> CoxeterGroups
> FiniteCoxeterGroups
> WeylGroups
> Lorenzian/HyperbolicCoxeterGroups: generic methods for Coxeter
> groups in an appropriate matrix representation
>
> - [ ] Allow <-1 coefficients in the Coxeter diagram
> - [ ] positive roots by depth
> - [X] Missing: elements of the group as matrices: for free from the
> above
> - [X] fundamental weights in the root space
> - [X] finding all reduced words
> - [ ] signature of the bilinear form
> - [ ] parabolic elements / elliptic elements of the group
> algo: build all elements, and select those by a criterion
> of diagonalizability and max modulus of the eigenvalues
> - [ ] Compute limit roots L.limit_roots(...)
> - [ ] Visualize limit roots L.plot_limit_roots()
New description:
Related tickets:
- #9290
- #13248
- #15703
- #15974
TODO:
- Cleanup this description!!!!!
* Do an experiment to produce a picture of limit roots.
- Shoot straight at producing a nice plot with a bunch of limit roots.
- Post the picture here.
- Benchmark the calculation.
- Every hack along the way is fair.
- Update the TODO list below with what would need to be done for a
proper implementation.
- Discard the experiment.
Matrix representation for Coxeter groups
- [ ] Root space for general Coxeter groups
sage: L = RootSystem(<"generalized" coxeter matrix>).root_space()
sage: W = L.reflection_group() returns an instance of
"CoxeterMatrixGroup"
- [ ] Generalize CoxeterMatrixGroup from WeylGroup
http://trac.sagemath.org/ticket/15703
Make it full featured
- [ ] Add a category for Lorenzian/HyperbolicCoxeterGroups
Inside the hierarchy:
CoxeterGroups
FiniteCoxeterGroups
WeylGroups
Lorenzian/HyperbolicCoxeterGroups: generic methods for Coxeter
groups in an appropriate matrix representation
- [ ] Allow <-1 coefficients in the Coxeter diagram
- [ ] positive roots by depth
- [X] Missing: elements of the group as matrices: for free from the
above
- [X] fundamental weights in the root space
- [X] finding all reduced words
- [ ] signature of the bilinear form
- [ ] parabolic elements / elliptic elements of the group
algo: build all elements, and select those by a criterion
of diagonalizability and max modulus of the eigenvalues
- [ ] Compute limit roots L.limit_roots(...)
- [ ] Visualize limit roots L.plot_limit_roots()
--
--
Ticket URL: <http://trac.sagemath.org/ticket/16087#comment:8>
Sage <http://www.sagemath.org>
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