Fusion algebras from Weyl Character Rings

[dwbump]

Fusion algebras for WZW conformal field theories can be computed easily as instances of WeylCharacterRings. The WeylCharacterRing code is modified with an optional parameter k, the level. If k==None the behavior is unchanged. However if k is a positive integer the corresponding fusion ring is created. The reason this works is that the Kac-Walton algorithm for computing the fusion products is closely similar to the Brauer-Klimyk (aka Racah-Speiser) algorithm that is already used by the WeylCharacterRing. One has only to add an affine reflection to make the algorithm compute the fusion product.

I tested this for level 2 in types A2 and B2, comparing with tabulated formulas in Feingold, Fusion Rules for affine Kac-Moody algebras.

A related patch is #15485.

CC: @tscrim @sagetrac-sage-combinat @dwbump

Component: combinatorics

Keywords: Fusion Ring, Verlinde Algebra

Author: Daniel Bump

Branch/Commit: 7546a0d

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/26440

[dwbump]

Commit: ed039ba

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New commits:

ed039ba

fusion algebras from weyl character rings

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Branch: public/fusion-26440

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

2b75fb5

contragredient or conjugation map

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cada3b3

don't import Matrix

[dwbump]

comment:5

For fusion rings, we should have a method that returns the (finite) canonical basis as a list or an iterator. Actually WeylCharacterRing has a basis method that it inherits from CombinatorialFreeModule.

To avoid confusion with this basis method, the method that returns the basis as a list could then be called fusion_basis.

So, what is the best way to enumerate all dominant weights of level <= k?

[tscrim]

comment:6

Replying to @dwbump:

For fusion rings, we should have a method that returns the (finite) canonical basis as a list or an iterator. Actually WeylCharacterRing has a basis method that it inherits from CombinatorialFreeModule.

To avoid confusion with this basis method, the method that returns the basis as a list could then be called fusion_basis.

I am not sure I like the basis for the fusion ring being anything other than what you want to call the fusion_basis. More generally, I am not sure I like that the basis for WeylCharacterRing is not the set of dominant integral weights. Since the fusion algebra is finite-dimensional, I definite would rather have basis correspond to the dominant weights of level <= k (which we would give in the __init__ to CombinatorialFreeModule).

So, what is the best way to enumerate all dominant weights of level <= k?

I just constructed this as a RecursivelyEnumeratedSet, but we could also use WeightedIntegerVectors. Between those two, I have not run timings to determine which is faster. Both should take 2 lines to construct (one to get the levels, one to construct the parent to iterate over).

[dwbump]

comment:7

I am not sure I like the basis for the fusion ring being anything other than what you want to call the fusion_basis.

Does this mean we should overwrite the existing basis method (which, you are pointing out is already not correct)?

I just constructed this as a RecursivelyEnumeratedSet, but we could also use WeightedIntegerVectors. Between those two, I have not run timings to determine which is faster. Both should take 2 lines to construct (one to get the levels, one to construct the parent to iterate over).

I doubt whether speed is an issue for this. The user will run it once. So you could choose on the basis of elegance rather than timing.

[tscrim]

comment:8

Replying to @dwbump:

I am not sure I like the basis for the fusion ring being anything other than what you want to call the fusion_basis.

Does this mean we should overwrite the existing basis method (which, you are pointing out is already not correct)?

Well, sort of. What I think should be done is the correct indices needs to be passed to CombinatorialFreeModule.__init__ (instead of self._space). This will also automatically put it into the category of FiniteDimensionalModulesWithBasis and give it extra methods.

With the existing basis method, the actual basis is too big as it says that, e.g., -\Lambda₁ is a basis elements. However, this is not allowed through more standard constructions. For simplicity of the implementation, this is nice. Although for the style='coroots', you can get a fair bit of speed by using weights expressed as linear combinations of fundamental weights rather than in the ambient basis (I have some really hacked together code to get some computations at a reasonable speed and memory usage I need in E₇ that I need to cleanup to get into Sage). Although it means some code duplication and added complexity. However, this makes it more ameable to the "correct" basis. Unless I am misunderstanding something about what this should be able to compute.

I just constructed this as a RecursivelyEnumeratedSet, but we could also use WeightedIntegerVectors. Between those two, I have not run timings to determine which is faster. Both should take 2 lines to construct (one to get the levels, one to construct the parent to iterate over).

I doubt whether speed is an issue for this. The user will run it once. So you could choose on the basis of elegance rather than timing.

Timing might matter if someone wants to construct a "large" rank and level example. In terms of elegance, I would say WeightedIntegerVectors is better and more inline with the output as sums of fundamental weights (as dense vectors).

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e8dec30

bugfix: reducible cartan types have no opposition involution

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cda3cce

fusion_basis method

[dwbump]

comment:11

I implemented fusion_basis method using RecursivelyEnumeratedSet. Travis, feel free to change basis per comments 6 and 8.

I also fixed a bug that broke WeylCharacterRing for reducible types.

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Changed work issues from The conjugation method needs to be created. to none

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Changed keywords from none to Fusion Ring, Verlinde Algebra

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0555717

FusionRing becomes a class

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d4166cc

names of basis elements can be customized

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6736b8c

fusion_weight element is made a method of WeylCharacterRing Elements and renamed highest_weight.

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d18b3d8

doctest for _dual_helper

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

102b00b

Added some doctets, fixed bug with some_elements, moved the refences to master ref file.

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[tscrim]

comment:25

I added some additional doctests, which led me to finding a bug with some_elements returning elements that are not basis elements of the fusion ring because the level of the fundamental weight is too large. I also moved the references to the masted reference file (and added a bit more details to them). If my changes are good, then you can set a positive review.

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Changed author from bump to Daniel Bump

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Reviewer: Travis Scrimshaw

[vbraun]

comment:28

Test failures (see patchbot)

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

768ca32

Test for reducibility rather than atomicness for dual.

[tscrim]

comment:30

The failures come from the difference between is_atomic and is_irreducible. I changed this to the appropriate tests. Please check.

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7546a0d

Test for reducibility rather than atomicness for dual.

[tscrim]

comment:32

I also did some small tweaks for speed.

[dwbump]

comment:33

This passes the tests in weyl_characters.py and I am also able to build the documentation. May I change the status back to positive_review?

[tscrim]

comment:34

Thank you. They also pass for me.

[vbraun]

Changed branch from public/fusion-26440 to 7546a0d