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16087 Limit roots in the geometric representation of Coxeter groups nthiery " 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.
- [X] Post the picture here.
- [X] 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, see #17798 and #16126
- [ ] 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()
" enhancement new major sage-6.4 group theory days57, plot, coxeter, days64 sage-combinat jipilab vripoll tscrim Jean-Philippe Labbé, Vivien Ripoll N/A #15703