Refactor Coxeter groups as matrix groups and non crystallographic root systems

[nthiery]

This is a follow up to #9290.

• Experiment with the infrastructure scales and benchmark

• CoxeterGraph, see Introduce a class for generalized Coxeter graphs #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 Create a class for Coxeter matrices and types #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
    Axioms and more functorial constructions #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.

Depends on #16120
Depends on #16126
Depends on #16130
Depends on #17798
Depends on #18152

CC: @sagetrac-sage-combinat @tscrim @jplab @sagetrac-vripoll @mathzeta

Component: combinatorics

Keywords: coxeter groups, days57

Author: Jean-Philippe Labbé, Vivien Ripoll

Branch/Commit: u/jipilab/refactor_coxeter_groups_as_matrix_groups_and_non_crystallographic_root_systems @ 09a1ff9

Reviewer: Nicolas M. Thiéry

Issue created by migration from https://trac.sagemath.org/ticket/15703

[fchapoton]

Changed keywords from none to coxeter

[nthiery]

Changed keywords from coxeter to coxeter groups, days57

[nthiery]

Author: Jean-Philippe Labbé, Vivien Ripoll

[nthiery]

Reviewer: Nicolas M. Thiéry

[tscrim]

comment:11

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).

[jplab]

comment:12

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...

[jplab]

Attachment: benchmark.png

[jplab]

Dependencies: 16120

[jplab]

Changed dependencies from 16120 to #16120

[sagetrac-vripoll]

Branch: u/vripoll/refactor_coxeter_groups_as_matrix_groups_and_non_crystallographic_root_systems

[jplab]

Changed dependencies from #16120 to #16120, #16126

[jplab]

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

[jplab]

Commit: 2846f67

[jplab]

Changed dependencies from #16120, #16126 to #16120, #16126, #16130

[jplab]

Attachment: benchmark2.png

[jplab]

Changed branch from u/vripoll/refactor_coxeter_groups_as_matrix_groups_and_non_crystallographic_root_systems to u/jipilab/refactor_coxeter_groups_as_matrix_groups_and_non_crystallographic_root_systems

[jplab]

Changed commit from 2846f67 to 09a1ff9

[jplab]

comment:21

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!!

---

New commits:

09a1ff9

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

[sagetrac-vbraun-spam]

comment:22

Attachment: test_rootsystems.sage.gz

[jplab]

Changed dependencies from #16120, #16126, #16130 to #16120, #16126, #16130, #17798, #18152