FusionRing enhancements

[dwbump]

The FusionRing, implemented in #26440 is the WZW fusion or Verlinde algebra. It needs certain methods to be useful: the twist or ribbon element, the S-matrix and the quantum dimension. This ticket is to implement these methods.

The ticket #29615 is related to this one.

CC: @dwbump @tscrim @sagetrac-sage-combinat @sagetrac-buciumas @willieab @physicstravels

Component: combinatorics

Keywords: FusionRing

Author: Guillermo Aboumrad, Daniel Bump

Branch/Commit: public/fusion_ring_operations-27036 @ 8918e84

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/27036

[dwbump]

Commit: 8735919

[dwbump]

Branch: public/fusion-27036

[dwbump]

New commits:

8735919

new methods for FusionRings: s-matrix, q_dimension, twists etc

[dwbump]

comment:3

Work issues:

• Further testing is needed. This should be able to duplicate the tables in arXiv:0712.1377: On classification of modular tensor categories. Eric Rowell, Richard Stong, Zhenghan Wang.
• Add a reference to that paper.
• Some methods need docstrings.
• s_matrix should return entries in a CyclotomicField, not CC.
• s_matrix element method should be kept but the code within the loop for i in rng: should be split off so the user can optionally work with an individual pair of elements rather than a matrix.
• Type E s-matrices seem conjugated. Figure out why.

Design decisions:

• Should the FusionRing be moved to a separate file? (perhaps not necessary unless code becomes much longer)
• What is the right way to make methods special to FusionRing not available to parent class WeylCharacterRing? (fixed)
• Can the ordered_basis, fusion_labels and basis methods be intelligently reorganized, and can we use python OrderedDict for the basis?

Figure out the right CyclotomicField which may be the square root of the one currently used.

[tscrim]

comment:4

There is the UniversalCyclotomicField if you do not want to be so careful about which roots you want.

As for making methods special to FusionRing not for the WCR, you should put those methods in the subclass.

[dwbump]

comment:5

Replying to @tscrim:

As for making methods special to FusionRing not for the WCR, you should put those methods in the subclass.

The original comment:3 is not clear but the issue is actually with WeylCharacterRing.Element methods.
that only pertain to FusionRing elements.

So I checked that the following works. We add a FusionRing.Element class that inherits from
WeylCharacterRing.Element. Then we move the WeylCharacterRing.Element methods such
as is_simple_obj,twists, etc. to the new class.

[embray]

comment:6

Retarging tickets optimistically to the next milestone. If you are responsible for this ticket (either its reporter or owner) and don't believe you are likely to complete this ticket before the next release (8.7) please retarget this ticket's milestone to sage-pending or sage-wishlist.

[sagetrac-git]

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

9cf678b

created FusionRing.Element class

[sagetrac-git]

Changed commit from 8735919 to 9cf678b

[sagetrac-git]

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

78007fa

doctests added, updated q_field, and S-matrix computation

[sagetrac-git]

Changed commit from 9cf678b to 78007fa

[sagetrac-git]

Changed commit from 78007fa to 2904d89

[sagetrac-git]

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

7a5b4ae	revised doctest
2904d89	work on revising self.s_ij

[embray]

comment:10

Ticket retargeted after milestone closed (if you don't believe this ticket is appropriate for the Sage 8.8 release please retarget manually)

[embray]

comment:11

As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).

[dwbump]

Changed commit from 2904d89 to 72c2c44

[dwbump]

comment:14

This patch is important but was neglected for a while. We are reviving it. The old branch was called public/fusion-27036 but this has been superceded by public/fusion_ring_operations-27036.

• The old patch was ported to python3 and recent development branch by physicstravels.
• The ordered_basis method has been abandoned in favor of a scheme based on the get_order and set_order methods of CombinatorialFreeModule.
• The method fusion_matrix was eliminated.

---

New commits:

16b56a8	new methods for FusionRings: s-matrix, q_dimension, twists etc
dd4f948	created FusionRing.Element class
eb09e3f	doctests added, updated q_field, and S-matrix computation
7be38ba	revised doctest
a5bc48f	work on revising self.s_ij
5438494	Update to py3. Fix doctests.
aabbb24	use get_order instead of ordered_basis
72c2c44	docstring edits

[dwbump]

Changed branch from public/fusion-27036 to public/fusion_ring_operations-27036

[sagetrac-git]

Changed commit from 04cefcf to c1338b9

[sagetrac-git]

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

c1338b9

doc cleanup

[sagetrac-git]

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

55b2e6b

added _test_verlinde method to check Verlinde formula

[sagetrac-git]

Changed commit from c1338b9 to 55b2e6b

[dwbump]

comment:51

The Verlinde formula is not quite stated in the usual way, because a complex conjugation is omitted, and N_ijk is used instead of Nk_ij. This appears to be correct (on the basis of testing) but why is it correct mathematically?

Since this isue came up I'm temporarily changing the ticket to Needs Work.

[dwbump]

Work Issues: Conjugation is omitted in the Verlinde formula. This is probably OK but needs to be justified.

[sagetrac-git]

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

0974b08

explain about conjugate in Verlinde formula

[sagetrac-git]

Changed commit from 55b2e6b to 0974b08

[sagetrac-git]

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

e9de384

revision of twists docstring

[sagetrac-git]

Changed commit from 0974b08 to e9de384

[dwbump]

Changed work issues from Conjugation is omitted in the Verlinde formula. This is probably OK but needs to be justified. to none

[dwbump]

comment:54

I added a bit more explanation about the Verlinde formula.

[sagetrac-git]

Changed commit from e9de384 to 0e97174

[sagetrac-git]

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

0e97174

fix newline before docstring

[sagetrac-git]

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

1b6e82f	Merge branch 'public/fusion_ring_operations-27036' of git://trac.sagemath.org/sage into public/fusion_ring_operations-27036
642e80f	Final reviewer changes.

[sagetrac-git]

Changed commit from 0e97174 to 642e80f

[tscrim]

Changed reviewer from tscrim to Travis Scrimshaw

[tscrim]

comment:57

I ended up changing the _test_verlinde to go through only some of the basis elements up to a limit in case someone wants to run the TestSuite on a very large dimensional fusion ring. This is close to what is done in _test_associativity. If you disagree, then I can revert this. If you are happy with the last change and tweaks, then you can set a positive review.

[sagetrac-git]

Changed commit from 642e80f to 8918e84

[sagetrac-git]

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

8918e84

remove S = tester.some_elements()

[mkoeppe]

comment:60

These have been merged into 9.2.beta4