#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: |
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Old description:
> Related tickets:
> - #9290
> - #13248
>
> 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
- #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()
--
Comment (by jipilab):
Hey Travis,
Ok, great! What you just said correspond for Cartan type to the "level 1"
definition for "generalized" Coxeter graphs introduced by Maxwell in '82
(Sphere packings and hyperbolic re
ection groups, J. Algebra 79 (1982), no. 1, 78-97.)
We will have a look at the patch!
--
Ticket URL: <http://trac.sagemath.org/ticket/16087#comment:7>
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