add rrule for flip

[sanderdemeyer]

This implements the rrule for the function flip().

[lkdvos]

Can you also add a test? See https://github.com/Jutho/TensorKit.jl/blob/ed09f138e41d8a69a703f6473e4acbb2838f2fb5/test/ad.jl#L224

[Jutho]

It turns out that this is not the right rrule, because flip is not an involution, i.e. it is not its own inverse. More specifically, the inverse of flip is flipping the index 3 times (for fermions). Probably we want to have an associated unflip function? I will make some changes to this PR to make this happen.

[lkdvos]

That sounds ... honestly scary 😬 It might be good to add a note of this in the docstring, since this is definitely something I would have wrongly assumed.

Possibly we could mimick the twist function and add an inv keyword?

[Jutho]

It essentially amounts to saying that flipping twice corresponds exactly to how you get the Frobenius-Schur indicator, which is ±1.

An inv keyword is probably the better option indeed.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 82.65%. Comparing base (ed09f13) to head (658d75a).

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #241      +/-   ##
==========================================
+ Coverage   82.61%   82.65%   +0.04%     
==========================================
  Files          43       43              
  Lines        5579     5593      +14     
==========================================
+ Hits         4609     4623      +14     
  Misses        970      970              

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[Jutho]

The failures are again related to ChainRulesTestUtils on Julia nightly (Julia 1.13). I checked at my own machine weather everything works on beta (Julia 1.12), and there everything seems fine, so I think this is good to go, unless there are further comments.

[lkdvos]

Left a small note about spelling, otherwise looks good to me!

[Jutho]

Great catch; I don't think we need another CI run for this so I will merge. (BTW: copilot completion also thought involutive, but I think that is wrong).