Opened 9 years ago

Last modified 5 years ago

#16087 new enhancement

Limit roots in the geometric representation of Coxeter groups — at Initial Version

Reported by: Nicolas M. Thiéry Owned by:
Priority: major Milestone: sage-6.4
Component: group theory Keywords: days57, plot, coxeter, days64
Cc: Sage Combinat CC user, Jean-Philippe Labbé, Vivien Ripoll, Travis Scrimshaw Merged in:
Authors: Jean-Philippe Labbé, Vivien Ripoll Reviewers:
Report Upstream: N/A Work issues:
Branch: Commit:
Dependencies: Stopgaps:

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  • 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?" Make it full featured

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()

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