`frobeniusschur` of a complex representation

[ogauthe]

Hi!

I was surprised to find that frobeniusschur(U1Irrep(1)) == 1. Since U1Irrep(1) is a complex irrep, I was expecting it to be 0 according to e.g Frobenius–Schur indicator definition. I can use dual(s) == s as a workaround, but the current definition is a surprise to me. Is this expected behavior?

I did not fully get the discussion in QuantumKitHub/TensorKit.jl#260, but it seems to focus on the ±1 case.

[Jutho]

Hi Olivier,

I guess it is true that the topological index is defined as ±1 only if a == dual(a) and as zero whenever a != dual(a). However, I am not sure if there is a direct mathematical formula that would explicitly produce the 0.

What we return with frobeniusschur has a proper mathematical definition that also works when a != dual(a), as explained in the docs:
https://quantumkithub.github.io/TensorKit.jl/stable/man/categories/#Duality-and-pivotal-structure

This notion of frobeniusschur will always be a complex phase (i.e. unit magnitude), but one that can freely be chosen for complex irreps.

I guess one can always write a new definition where

topologicalfrobeniusschur(a) = (a == dual(a)) ? frobeniusschur(a) : 0.

[ogauthe]

Thank you for your answer.

In group theory the convention is to define the Frobenius–Schur indicator as $\int_{g\in G} \chi(g^2) d\mu$, where $\chi$ is the character, and the formula returns 0 for a complex irrep. It seems to me you define a slightly different concept here and I got confused with the denomination, but I guess this formula only makes sense for groups.

I can make a PR to define topologicalfrobeniusschur and update the doc to mention this.

[Jutho]

I am not sure if topologicalfrobeniusschur is really a good name; that was just to make my point clear. Maybe we should revisit the name of what we currently call frobeniusschur and see if there is a better name for that?

[Jutho]

This is from the famous Kitaev Appendix E (from https://arxiv.org/pdf/cond-mat/0506438)

[ogauthe]

After some research, "Frobenius-Schur" returns mostly results about group theory. Searching explicitly for generalizations to category theory, I found https://scipost.org/preprints/scipost_202209_00037v1/. It does not give a name to the concept in the a != dual(a) case, then defines Frobenius-Schur only in the self-dual case. This is consistent with Kitaev's definition.

Therefore I would vote for frobeniuschur to return values -1, 0, 1 and use a different name for the current concept. I appreciate changing the name of the current function is a breaking change with all the annoying consequences.

[Jutho]

Yes, though we are discussing other breaking changes as well. For this one, the problem is mostly lack of inspiration for a good name. Yet another alternative is to keep frobeniusschur, and introduce frobeniusschurindicator for the -1, 0, 1 version, though this may be a bit too confusing. Wouldn't mind hearing @lkdvos' input as well.

[lkdvos]

I don't feel too strongly about this one. I'm less aware of people using this at the moment for specific purposes, and additionally I'm not too aware of many specializations for this function, so I think we are mostly just using this as a specialization point for more efficient implementations that avoid evaluating Fsymbol. If we are anyways going for a breaking change, this wouldn't be too annoying to include, although it would be nice to come up with a name that isn't too far off given that we do mention this in the paper

[ogauthe]

I can think of an explicit like flip_to_dual_phase, although this is quite different from the current name.

[Jutho]

Or simply frobeniusschurphase and frobeniusschurindicator? I am curious to know what you will use the indicator for?

[lkdvos]

How about we keep frobeniusschur, add frobeniusschurindicator for the (-1, 0, 1) version, and update the relevant parts of the docs to clearly state these differences? That would be non-breaking, and still provide the functionality you want?

[ogauthe]

I am curious to know what you will use the indicator for?

I am working on combining bra-ket symmetry with standard symmetry sectors. It turns out there are three different cases to consider, which correspond to the 3 possible values of the Frobenius-Schur indicator.

I do not really mind between frobeniusschur or frobeniusschurindicator. I find it confusing to call the current phase with a name associated with Frobenius-Schur since the result does not match its common group theory definition. I appreciate the concept is related, so I guess frobeniusschurphase is acceptable.

[Jutho]

I am fine with keeping frobeniusschur, but as we are considering a much more breaking change in TensorKitSectors.jl (where this issue really belongs), I don't think this is such an important argument to consider at this particular point in time. So I think now is also the time to get this one right and then bundle both breaking changes in a new breaking version.

Also, frobeniusschur is a bit of a lazy name in that it is just two person names with the actual noun/concept missing. I am not necessarily very satisfied with frobeniusschurphase for our concept, but it is the best I could come up with so far.