Improvements to root systems

[tscrim]

Implement various things for affine root systems:

• The symmetric form ( | ).
• Methods for checking the short/long/imaginary roots.
• Modelling the positive roots roots in affine types.
• Basic untwisted types (so type A_2n⁽²⁾ would give A_2n).
• (Dual) Coxeter numbers for finite types.

[tscrim]

New commits:

​0fa598a	imported patch root_system_data-ts.patch
​870837f	imported patch root_systems-ts.patch

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e40f892	Changed horizontal to basic_untwisted.
​930782b	Fixed indentation error.
​f87789d	Merge branch 'master' into public/combinat/root_systems/improvements-15384

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b4c7865

Merge branch 'master' into public/combinat/root_systems/improvements-15384

[git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

​6100a4b	Merge branch 'public/combinat/root_systems/improvements-15384' of trac.sagemath.org:sage into public/combinat/root_systems/improvements-15384
​a5a6b1c	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​f802ae1	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​d83982c	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​ecf8f76	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​c52545c	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​14c367e	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​fe68bf6	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​896e682	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​dd976c7	Merge branch 'develop' into public/combinat/root_systems/improvements-15384

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2c7f793

Merge branch 'develop' into public/combinat/root_systems/improvements-15384

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ea1641f	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​054b37c	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​30de9e8	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​122ca6e	Merge branch 'develop' into public/combinat/root_systems/improvements-15384

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​87cf9ed	Merge branch 'public/combinat/root_systems/improvements-15384' of trac.sagemath.org:sage into public/combinat/root_systems/improvements-15384
​c5f9747	Added placeholder for symmetric_form in WLR.
​64ac7a5	Fixed doctests.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​1817877

Almost there.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​18fd094

Fixed affine weight lattice symmetric form.

[tscrim]

Hey Dan,

I've fixed it. The (first r) columns are given by P.simple_roots(), and it's the inverse matrix should have the Kac labels (which it does).

[bump]

Glad you got it fixed. I'll start looking at the patch.

[nthiery]

Travis: do you mind updating the description of the ticket to give an overview of the changes?

Thanks!

[bump]

The scalar product that was implemented is fundamental in Kac' book and for that alone this is an important patch.

I got one doctest failure in weight_space.py:

sage -t weight_space.py
**********************************************************************
File "weight_space.py", line 131, in sage.combinat.root_system.weight_space.WeightSpace
Failed example:
    for ct in CartanType.samples(crystallographic=True)+[CartanType(["A",2],["C",5,1])]:
          TestSuite(ct.root_system().weight_lattice()).run()
          TestSuite(ct.root_system().weight_space()).run()
Expected nothing
Got:
    Failure in _test_not_implemented_methods:
    Traceback (most recent call last):
    [snip]
    AssertionError: Not implemented method: _symmetric_form_matrix

In cartan_type.py you could delete the sentence on line 108 since you do implement Coxeter numbers!

For the positive roots, imaginary roots, etc. The user might like to know how to get a list of them. For *untwisted* types you can do this.

sage: PR=RootSystem(['A',3,1]).root_lattice().positive_real_roots()
sage: [PR.unrank(i) for i in [0..4]]
[alpha[1], alpha[2], alpha[3], alpha[1] + alpha[2], alpha[2] + alpha[3]]

Is there a more obvious way? And should something like this be in the doctest?

In the method basic_imaginary_roots the docstring still refers to simple imaginary roots (twice).

It would be good if there were pointers to Kac' book in some places. For example, in symmetric_form, the doc could have a pointer to Chapter 6. The Coxeter number method could have a pointer to Bourbaki, Lie Groups and Lie Algebras V.6.1.

[tscrim]

I've created #16320 to make it easier to access elements in an infinite list (but I have not merged it in so it's not a dependency). I'm not sure if at the end of the day we want to keep using DisjointUnionEnumeratedSets and possibly adding better general support for those or making our own special subclasses and having them be better targeted for the roots here. Nicolas, do you have any thoughts/comments on this?

IMO this can be left for a follow-up ticket unless there is a simple and clear better solution. Although if #16320 is a good way to go, I'll merge that in and update the doctests accordingly.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​cb88df7	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​760194b	More changes and fixes.

[bump]

Still two doctest failures in root_lattice_realizations.py.

sage -t root_lattice_realizations.py
**********************************************************************
File "root_lattice_realizations.py", line 634, in sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods.positive_real_roots
Failed example:
    PRR = L.positive_real_roots(); PRR
Expected nothing
Got:
    Positive real roots of type ['A', 3, 1]
**********************************************************************
File "root_lattice_realizations.py", line 635, in sage.combinat.root_system.root_lattice_realizations.RootLatticeRealizations.ParentMethods.positive_real_roots
Failed example:
    [PRR.unrank(i) for i in range(10)]
Expected:
    Positive real roots of type ['A', 3, 1]
    [alpha[1],

   -snip-

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​eab1afe

Fixed doctest.

[tscrim]

Whoops, forgot to commit.

[bump]

All tests passed!

The following breaks:

[RootSystem?(['A',2,2]).root_lattice().positive_roots().unrank(x) for x in [0..10]]

It works for untwisted types. Is this a limitation of the patch or an unexpected failure?

If it is a limitation of the patch maybe it should be noted in the patch description. Also it might be better to fail with a not implemented error than with:

TypeError: unsupported operand type(s) for //: 'RootSpace_with_category.element_class' and 'int'

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​66f5674

Fixed type BC positive roots.

[tscrim]

That was an error. Fixed now.

[bump]

What about this?

sage: RootSystem(['D',4,2]).root_lattice().positive_roots()
---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-11-ea364fc7aada> in <module>()
----> 1 RootSystem(['D',Integer(4),Integer(2)]).root_lattice().positive_roots()

-snip -

AttributeError: 'NoneType' object has no attribute 'dual'

BTW Sage reports this root system as Root lattice of the Root system of type ['C', 3, 1]^*

If I have my facts straight Macdonald introduced a classification of affine root systems in his paper "Affine root systems and the Dedekind eta function" Inventiones 1972 but Kac didn't follow Macdonald. He introduced his own classification. So we are entering root systems in Kac' notation and they are getting converted automatically to Macdonald's notation. Of course that policy was not introduced by this patch but I mention that I've personally found it a little confusing.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​e17ce53	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​7e83910	Fixes for other twisted root systems.

[tscrim]

Another (stupid) error on my part. Fixed.

Also if you want to display in Kac's notation, you can do the following (it's somewhat hidden at the bottom of CatanType?):

sage: CartanType.global_options(notation='Kac')
sage: CartanType(['D',4,2])
['D', 4, 2]

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​c91754d	Merge branch 'develop' into public/combinat/root_systems/improvements-15384
​82c6257	Fixed doctest.
​1ffb4bb	Fixed doctest and for the weight lattice.

[bump]

The following returns an error:

sage: sr=RootSystem(['A',6,2]).weight_lattice().simple_roots()
sage: sr[0].symmetric_form(sr[1])
...
TypeError: no conversion of this rational to integer

There is no error if we use the root_lattice.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​bccea25

Worked around a minor coercion issue and added doctests.

[tscrim]

Hey Dan,

I've fixed the issue; the matrix stack() command doesn't coerce (well, it's more of I've worked around it and is now #16399). I've added some more doctests about this. I'll merge in the latest beta later tonight.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​f4cee52

Merge branch 'develop' into public/combinat/root_systems/improvements-15384

[bump]

This patch implements some important things from Chapter 6 of Kac' book for both twisted and untwisted affine Lie algebras, as methods of the root_lattice_realization or the weight_lattice_realization. These include a method of enumerating the positive roots (both real and imaginary), the invariant bilinear form, and a few other things. I tested with Kac' book in front of me, and all the problems I found have been addressed.

[tscrim]

Thank you for doing the review Dan.

[vbraun]

Doctests fail

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​7285940

Fixed bad merging in simple_roots_tilde.

[tscrim]

Bad merging on my part; ended up overwriting simple_roots_tilde().