Refactor Coxeter groups as matrix groups and non crystallographic root systems

[Nicolas M. Thiéry]

This is a follow up to #9290.

• Experiment with the infrastructure scales and benchmark

• CoxeterGraph, see #16126
  • Create a class similar to DynkinDiagram Starter: an edge-labeled graph.
  • Edge labels: m_{i,j}, possibly with number <-1 for oo
  • method dynkin_diagram() which builds the cartan matrix for the geometric representation Starter: just make this a function

• Update DynkinDiagram to support non crystallographic case:
  • Add an argument base_ring to the constructor
  • Add a method base_ring
  • Make add_edge honor this method when automatically adding edges
  • Update cartan_matrix() to use the base_ring
  • Add a method _test_base_ring that checks that all edge labels are indeed in this base ring
  • Implement is_crystallographic testing if the base ring is ZZ
  • Add an argument symmetric=False to the constructor, and make add_edge and symmetrizer use it.
  • Add a method _test_dynkin_diagram that tests that the Dynkin diagram indeed defines a proper root system. See in particular cartan_matrix.is_generalized_cartan_matrix.
  • adapt column() and row() method to give the labels in the base ring

• Update CartanMatrix, see #17798
  • Add a base ring argument to the constructor
  • Update is_crystallographic
  • Update is_affine
  • Update is_finite
  • Update is_generalized_cartan_matrix

• CartanType
  • Possibly update to accept appropriate data to build a CoxeterGraph (e.g. a matrix)
  • Add a base_ring method?
  • Decide on the semantic of is_crystallographic (symmetrizable or not?), and if possibly add an is_... method to decide whether the entries are integral or not.
  • Provide a dynkin_diagram method that builds the Dynkin diagram from the Coxeter diagram when available
  • Test: H_3 and friends should have a working dynkin_diagram method

• RootSystem
  • Decide on the meaning of root_lattice: either disable it in the non integral case, or have it be the span of the roots over the smallest available ring.

• RootLatticeRealizations:
  • Feed this to RootSystem, and check that the root space and weight space are built properly.
  • Rename the weyl_group method to reflection_group, with an alias from weyl_group; update the setting of the category.
  • Define a new projection "transversal" to visualise root systems (and find a right name for it)
  • Long run: stuff specific to the crystallographic case, starting with this weyl_group method, should go in RootLatticeRealizations.Crystallographic. That's for a follow up ticket on using axioms for root systems; but let's not depend on #10963 right now.

• RootSpace (for this ticket or some follow up):
  • Define the inner product
  • adapt the is_positive_root to make it work for any base ring
  • Signature of the bilinear form

• CoxeterMatrixGroup and WeylGroup:
  • Refactor WeylGroup to make it a subclass of CoxeterMatrixGroup, and lift as many features as possible from WeylGroup to CoxeterMatrixGroup.
  • Check that, with a proper Dynkin diagram, the conversion to GAP issue does not appear
  • Now or later: we probably want the Weyl group elements to be represented by Sage matrices, but keep a handle to the corresponding Gap group. Currently one has to make a choice between !MatrixGroup_generic and !MatrixGroup_gap.

• Plotting:
  • add a family_of_points method in the projections to be used by the "transversal projection"

• Update WeylGroups:
  • inversions: use the "root_lattice" by default?

Tests:

  sage: C = CoxeterDiagram(...)           # good name? or CartanDatum(coxeter_matrix=...) [1] ? or?
  sage: L = RootSystem(C).root_space()
  sage: W = L.reflection_group()
  sage: W = CoxeterGroup(['H',3])
  sage: W.domain()

Sage Days 57 in Cernay will be a good occasion to work on this.

Follow ups: #16087

[1]: Generally speaking, it's planned to rename CartanType to CartanDatum.

[Travis Scrimshaw]

I believe I'm taking care of the inner product on the root space in #15384 (which I called symmetric_form()). Also for a followup ticket, we should implement/refactor things for symmetrizable and the non-symmetrizable types (for when we get the hyperbolic types done).

[Jean-Philippe Labbé]

Very good!

Just a small suggestion: I would call the function "bilinear_form". Although it is true that we deal with symmetric forms so far...

[Jean-Philippe Labbé]

New commits:

​a2fdd20	First draft with some hacks to make root systems visualization work
​d025f0f	Added a __float__ method to the class Universal Cyclotomic Field
​2846f67	Merge branch 'ticket16120' into t/15703/refactor_coxeter_groups_as_matrix_groups_and_non_crystallographic_root_systems

[Jean-Philippe Labbé]

I adapted the TODO list in relation with the latest changes I just pushed. The script joined allows you to create the pictures and do some tests...

Now we have to work!!

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New commits:

​09a1ff9

First dirty version to get a TODO list for the ticket 15703