add AbstractGroupElement and ZNElement

[Jutho]

Build the foundation for adding sectors corresponding to group elements, in order to use $\mathrm{Vec}_{\mathsf{G}}^\omega$ categories, and explicitly implement the case of $\mathbb{Z}_N$, which will be useful for testing purposes of UniqueFusion sectors with nontrivial Fsymbol values.

Potentially of interest to @lalooten, @Chenqitrg , @borisdevos .

[codecov]

Codecov Report

❌ Patch coverage is 79.41176% with 14 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/groupelements.jl	78.12%	14 Missing ⚠️

Files with missing lines	Coverage Δ
src/TensorKitSectors.jl	16.66% <ø> (ø)
src/irreps.jl	95.43% <100.00%> (ø)
src/sectors.jl	89.24% <100.00%> (ø)
src/groupelements.jl	78.12% <78.12%> (ø)

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[lkdvos]

Left a few comments, but overall looks good to me! Could probably do with some minor reformatting and making sure the docstrings are all up to date though.

[Jutho]

I will update the PR tonight, as I still have to install the Runic formatter in my VSCode at the office.

[Jutho]

Ok, with some delay, here are the updates. I also figured out and implemented the correct braiding rules for the ZNElement{p} case, which in the end was quite simple. It took me more time to discover that there was actually an error (not that it matters) in the hexagon_equations, at least according to the convention for Rsymbol that we use in the documentation and manual. Rsymbol(a, b, c) is the coefficient you get when you apply a braid to a splitting tensor with first (left) output a and second (right) output b, and coupled sector c, i.e. when denoting the fusiontensor objects as X(a, b, c), the relation is

$$\tau_{1, 2} \circ X(a, b, c) = R(a, b, c) X(b, a, c)$$

But this was not how it was used in the hexagon_equations, i.e. a and b were reversed. However, for all the sectors we currently test, this does not matter.

[Jutho]

I don't understand the JET error, nor why it only happens on Julia latest on Mac and Ubuntu.

[lkdvos]

Looks more or less good to go for me, I also can't say I understand the JET error. I'm wondering if it is complaining that no type_repr implementation exists for the abstract group element supertype?

[lkdvos]

In principle we could specialize the cocycle and Rsymbol for this case to return $\pm1$ as Int if I'm not mistaken, but this might not really be worth it

[Jutho]

Yes, I had this at first. But it seems that also for SU2Irrep we return ±1.0 so I opted for floating point. Of course, in that case the Fsymbol definitely needs floating point numbers, so yes, the N ==2 && p == 1 case could use integers for both Fsymbol (cocycle) and Rsymbol.

[Jutho]

Looks more or less good to go for me, I also can't say I understand the JET error. I'm wondering if it is complaining that no type_repr implementation exists for the abstract group element supertype?

That's also what I think. Not sure if there is a better or cleaner way to do this.

[Jutho]

This is very weird, the latest commit, which was just a code suggestion in a doc string, completely changed the CI results, again because of the JET error. That doesn't seem right.

[Jutho]

Ok, actually the cocycles were having an incorrect factor $1/N$ instead of $1/N^2$, which made them all trivial. This is now correct, and also makes that the N==2 && p==1 case is no longer fermionic, but anyonic (twist = i). This is what I indeed thought, because that should be the semion.

[kshyatt]

FWIW the Jet error doesn't occur for me just now with Julias 1.10 or 1.11 on OSX. Let me try rebasing this and we'll see how CI does...

[lkdvos]

I guess we still don't have any idea what's going on here? Any ideas on circumventing the issue?

[kshyatt]

I have one, do you mind if I try some CI based debugging? I can't replicate this locally on macOS or Linux...

[lkdvos]

Of course, go ahead!

[kshyatt]

This fixed the problem for me locally on a linux box, 🙏

[borisdevos]

Same here on Windows!