Add documentation for integrable representation (#15806) to Lie methods thematic tutorial

[bump]

The IntegrableRepresentation? code for working with integrable representations of affine Lie algebras is documented in the reference manual but not in the Lie Methods and Related Combinatorics thematic tutorial. This ticket is to add documentation to the thematic tutorial.

[bump]

New commits:

​506f2ec

Lie thematic tutorial revisions

[git]

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

​874e575	writing about weight lattice, integrable representations, etc.
​1e8dadf	minor edits

[git]

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

​73afeaf

minor fix

[git]

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

​1903175

mention methods of weight lattice

[git]

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​6b85a92

some writing

[git]

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

​d54d0f8

some edits

[git]

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

​440d8e2

Coxeter number

[kdilks]

Once this is done, I can look at it for typos, and see how it looks from the perspective of somebody that's not an expert on this stuff. Would probably still need an expert to at least glance over it to make sure nothing is majorly wrong.

[aschilling]

I would be glad to help review this ticket. Dan, were there sections you wanted me to write? Why have some titles changed from capitals to lower case letters?

[tscrim]

I am also willing to help with either writing or reviewing too.

[git]

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

​d9066be

reorganization: moved weight rings section and created a new section for integrable representations

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​83a5de7

integrable.rst

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​06d9be6

doc edits and a new method in integrable_representations.py

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​cbebbd1

minor fixes

[git]

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

​9ed2004

mult in Integrable example

[git]

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

​943d20e

added mu option to level method in IntegrableRepresentation

[git]

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

​04e3f60

string functions

[tscrim]

I would not have level take an optional weight in the IntegrableRepresentation. Weights already know about their level:

sage: La = RootSystem(['A',3,1]).weight_lattice().fundamental_weights()
sage: (La[2]+La[0]).level()
2
sage: al = RootSystem(['A',3,1]).root_lattice().simple_roots()
sage: al[1].level()
0
sage: La = RootSystem(['A',3,1]).ambient_space().fundamental_weights()
sage: La[1].level()
1

[git]

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

​cef780a

integrable.rst

[git]

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​d595425

changes found by khlee

[git]

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

​f40bb29

more about modular characteristics

[git]

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

​240c8a8

correction

[git]

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

​0261f99

finished 2Lambda[0] example

[git]

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

​07c9754

added some examples from khlee

[bump]

A link to the built documentation may be found at:

​http://sporadic.stanford.edu/thematic_tutorials/lie.html

The new sections are called "Affine Root System Basics" and "Integrable Highest Weight Representations of Affine Lie algebras".

[aschilling]

I have trouble checking out the branch. It says that the apply fails. Also the branch is currently red.

Some minor comments in the meantime:

• 'lie Group Basics' should be 'Lie Group Basics' in the index
• 'infinity crystals' should be 'Infinity crystals' in the index

[aschilling]

Since I cannot check out the branch, I cannot make these trivial changes. But here are some more typos:

• othrwise -> otherwise in the section "Untwisted Affine Kac-Moody Lie Algebras"
• I would use \ldots rather than \cdots in lists
• alpha_1 should be \alpha_1 in the section "Twisted Types"
• spce -> space in the section "Roots and Weights"
• The following sentence "(For the twisted cases, see [Kac]" seems unfinished or a bracket is missing.
• The link ~sage.combinat.root_system.extended_affine_weyl_group seems broken.

[git]

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

​55577f7

Merge branch 'public/tutorial-25937' of git://trac.sagemath.org/sage into public/tutorial-25937

[tscrim]

I've merged in 8.4.beta0 (which just had a trivial conflict).

[bump]

Replying to aschilling:

What happens when you try to check out the branch? I can check it out after first pulling from the develop branch, then:

git checkout origin/public/tutorial-25937 -b tutorial-25937

[bump]

Work issues:

Anne and Travis have some comments that need to be addressed. I'll try to do those now.

Should the section on affine crystals have any references back to this section? Specifically I discuss the question of extended versus not extended root lattice.

We had discussed adding material about Kac and BKK crystals in a new section. That could be another ticket or it could be part of this one.

I have trouble making the backward link to the reference manual on extended affine Weyl groups work at the very end of affine.rst. S similar link to the reference manual for IntegrableRepresentation? class works fine.

[git]

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

​73986ab

revert level method of IntegrableRepresentations

[aschilling]

Replying to bump:

Replying to aschilling:

What happens when you try to check out the branch? I can check it out after first pulling from the develop branch, then:

git checkout origin/public/tutorial-25937 -b tutorial-25937

Travis rebased the branch. Now I can check it out. Before that did not work for me.

[khlee]

Here are some corrections of typos and comments on "Integrable Highest..."

---

second paragraph: weight decomposition --> weight space decomposition

--The support of an integrable highest weight representation

first and second paragraphs: Both \Lambda and \lambda are used. We need to choose one. For consistency, \Lambda is better.

third paragraph: a increasing sequence --> an increasing sequence

---Modular Forms

first paragraph: weakly modular form --> weakly holomorphic modular form

I think that "weakly holomorphic" is more commonly used.

first paragraph: weakly modular means --> weakly holomorphic means

third paragraph: weakly modular form --> weakly holomorphic modular form

third paragraph: the number of positive roots of $\mathfrak g^{\circ$ --> the number of positive roots plus half the rank of $\mathfrak g}\circ$

---Sage methods for integrable representations

first paragraph: highest weight Lie algebras --> highest weight representations

second paragraph: two dominant multiple weights --> two dominant maximal weights

second paragraph: Since $s_0$ fixes $2 \Lambda_1 -\delta$, it is not a good example. The following Sage code does not show the action of the Weyl group element. It may be a good idea to remove the part "We apply...the corresponding dominant maximal weight" as this point will be illustrated in the next paragraph.

fourth paragraph: Again, the choice of $s_0$ is not good. Any other choice will work.

sixth paragraph: the parentheses in the denominator of the first product is broken.

sixth paragraph: the factor $q^{{-1/8}$ --> the factor $q}{1/8}$

seventh(?) paragraph: The infinite sums in the parentheses for the string functions need to begin $1$ instead of $q$.

seventh(?) paragraph: both weakly modular forms --> both weakly holomorphic modular forms

seventh(?) paragraph: two is als a weakly modular form--> two is also a weakly holomorphic modular form

A period is missing after [KimLeeOh2017].

[aschilling]

Replying to bump:

Work issues:

Anne and Travis have some comments that need to be addressed. I'll try to do those now.

Should the section on affine crystals have any references back to this section?

Yes, that would be good!

We had discussed adding material about Kac and BKK crystals in a new section. That could be another ticket or it could be part of this one.

Let us make that another ticket unless you want to write these sections now.

[git]

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

​1e1802b

addressed comments by Travis, Anne and Kyu-Hwan

[tscrim]

Replying to aschilling:

Replying to bump:

We had discussed adding material about Kac and BKK crystals in a new section. That could be another ticket or it could be part of this one.

Let us make that another ticket unless you want to write these sections now.

I think it would be better to have that as a second ticket anyways (better granularity of purpose on each ticket).

[git]

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

​9282d8e	Fixes to mult and added a few more doctests.
​9099c2e	Fixes for formatting, typos, and conventions.

[tscrim]

Thank you again Dan for your work on this and improving Sage's documentation/tutorials as a learning resource.

I have made a bunch of small changes in the tutorials. Most of them are small formatting changes and typo fixes, but for a few of my changes, I added a little more detail or did some rewording.

I am happy with the current state.

[bump]

Can anyone figure out how to make the link at the end of the section on Extended Affine Weyl Groups work?

There is a similar link to the reference manual for the IntegrableRepresentation? class and that works.

The broken link is the very last thing here: ​http://sporadic.stanford.edu/thematic_tutorials/lie/affine.html

[tscrim]

Did you try :mod:`~sage.combinat....`?

I can build the docs, but I can't look at the compiled version until I get to my desktop on Monday.

[tscrim]

I just realized that I missed something with formatting in integrable.rst (line 269):

 .. MATH::
 
     \begin{aligned}
-    c^{2\Lambda_0}_{2\Lambda_0} & = q^{-1/16}(1+q+3q^2+5q^3+10q^4+16q^5+\cdots),
+    c^{2\Lambda_0}_{2\Lambda_0} & = q^{-1/16}(1+q+3q^2+5q^3+10q^4+16q^5+\cdots), \\
     c^{2\Lambda_0}_{2\Lambda_1-\delta} & = q^{7/16}(1+2q+4q^2+7q^3+13q^4+21q^5+\cdots).
     \end{aligned}

I can push this change if you want.

[tscrim]

In addition to A001700, it may also be worthwhile mentioning the doubled version: A128015. Thoughts?

[git]

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

​61c8992

3 comments by Travis

[bump]

Did you try :mod:~sage.combinat....?

That works. Thanks!

I implemented the last 3 comments from Travis and pushed.

---

New commits:

​61c8992

3 comments by Travis

[tscrim]

No problem.

Anyone else have any comments? (Kevin?) I am ready to set this to a positive review.

[khlee]

I have read the document one more time and found a few typos. After correcting them, I would agree to set this to a positive review. Thanks!!

---

--The support of an integrable highest weight representation

first paragraph: $L(\lambda)$ --> $L(\Lambda)$ (There are two occurrences of this.)

---Sage methods for integrable representations

first paragraph: A period is missing at the end of the last sentence.

second paragraph: The sentence "Then we compute the string function at this weight, which we see agrees with the string function for the corresponding dominant maximal weight:" seems out of place. Could we delete it?

A period is missing at the end of the displayed math in the sentence begin with "Reading what the OEIS..."

The formulas of two string functions are not adequately aligned.

---

[git]

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

​b047996

comments by khlee

[bump]

I made Kyu-Hwan's corrections except this one:

The formulas of two string functions are not adequately aligned.

I don't see the alignment problem.

---

New commits:

​b047996

comments by khlee

[aschilling]

In the introduction to "Integrable Highest Weight Representations of Affine Lie algebras", there is still a missing link:

weyl_character_ring

[aschilling]

The link to the last OESIS 'OEIS sequence 128015' seems broken. Other than these last two link problems, I am also happy to set a positive review.

[kdilks]

@Travis it seems like enough other people have already gotten a chance to read through it.

[git]

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

​cb48b6c

two broken links found by Anne

[khlee]

I am not sure what happened to the alignment problem. In any case, it disappeared, and I am happy with the current version.

[git]

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

​adbc210

two more typos

[vbraun]

Should be marked as optional internet:

File "src/doc/en/thematic_tutorials/lie/integrable.rst", line 224, in doc.en.thematic_tutorials.lie.integrable
Failed example:
    [oeis(x) for x in V.strings().values()]
Exception raised:
    Traceback (most recent call last):
      File "/Users/buildslave-sage/slave/sage_git/build/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 614, in _run
        self.compile_and_execute(example, compiler, test.globs)
      File "/Users/buildslave-sage/slave/sage_git/build/local/lib/python2.7/site-packages/sage/doctest/forker.py", line 1025, in compile_and_execute
        exec(compiled, globs)
      File "<doctest doc.en.thematic_tutorials.lie.integrable[12]>", line 1, in <module>
        [oeis(x) for x in V.strings().values()]
      File "sage/misc/lazy_import.pyx", line 354, in sage.misc.lazy_import.LazyImport.__call__ (build/cythonized/sage/misc/lazy_import.c:3759)
        return self.get_object()(*args, **kwds)
      File "/Users/buildslave-sage/slave/sage_git/build/local/lib/python2.7/site-packages/sage/databases/oeis.py", line 382, in __call__
        return self.find_by_subsequence(query, max_results, first_result)
      File "/Users/buildslave-sage/slave/sage_git/build/local/lib/python2.7/site-packages/sage/databases/oeis.py", line 510, in find_by_subsequence
        return self.find_by_description(subsequence, max_results, first_result)
      File "/Users/buildslave-sage/slave/sage_git/build/local/lib/python2.7/site-packages/sage/databases/oeis.py", line 475, in find_by_description
        sequence_list = _fetch(url).split('\n\n')[2:-1]
      File "/Users/buildslave-sage/slave/sage_git/build/local/lib/python2.7/site-packages/sage/databases/oeis.py", line 195, in _fetch
        raise IOError("%s\nError fetching %s." % (msg, url))
    IOError: <urlopen error [Errno 60] Operation timed out>
    Error fetching http://oeis.org/search?q=1%2C+1%2C+3%2C+5%2C+10%2C+16%2C+28%2C+43%2C+70%2C+105%2C+161%2C+236&start=0&fmt=text&n=3.
**********************************************************************
1 item had failures:
   1 of  32 in doc.en.thematic_tutorials.lie.integrable
    [31 tests, 1 failure, 131.65 s]
----------------------------------------------------------------------
sage -t --long src/doc/en/thematic_tutorials/lie/integrable.rst  # 1 doctest failed
---------------------------------------------------------------------

[git]

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

​a9119bb

oeis test is optional internet

[bump]

This had a positive review. Per comment:60 I changed the oeis test to optional -- internet. I was unsure whether it is appropriate for me to put the positive review back so I changed it to needs review.