Fix apply_diff_group for some actions

[mateuszbaran]

I've also added differential of matrix inversion.

TODO:

• Check if this also affects translate_diff, inverse_translate_diff and adjoint variants.
• Add tests & check coverage.

[codecov]

Codecov Report

Merging #673 (0ad3fad) into master (f9a6b19) will increase coverage by 0.16%.
The diff coverage is 100.00%.

❗ Current head 0ad3fad differs from pull request most recent head 6714cff. Consider uploading reports for the commit 6714cff to get more accurate results

@@            Coverage Diff             @@
##           master     #673      +/-   ##
==========================================
+ Coverage   99.41%   99.57%   +0.16%     
==========================================
  Files         108      108              
  Lines       10600    10675      +75     
==========================================
+ Hits        10538    10630      +92     
+ Misses         62       45      -17     

Files	Coverage Δ
ext/ManifoldsTestExt/tests_general.jl	99.15% <ø> (+3.23%)	⬆️
ext/ManifoldsTestExt/tests_group.jl	100.00% <100.00%> (+0.26%)	⬆️
src/Manifolds.jl	86.66% <ø> (ø)
src/groups/addition_operation.jl	100.00% <100.00%> (ø)
src/groups/general_unitary_groups.jl	100.00% <100.00%> (ø)
src/groups/group.jl	100.00% <100.00%> (ø)
src/groups/group_operation_action.jl	100.00% <100.00%> (ø)
src/groups/multiplication_operation.jl	100.00% <100.00%> (ø)
src/groups/power_group.jl	100.00% <100.00%> (ø)
src/groups/product_group.jl	100.00% <100.00%> (ø)
... and 1 more

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[mateuszbaran]

Sorry that it takes a while but it's fairly convoluted. @olivierverdier if I'm on the right track, it looks like our differentials are incorrect exactly up to sign when differentiating wrt. something taken with an inverse.

[olivierverdier]

Sorry that it takes a while but it's fairly convoluted. @olivierverdier if I'm on the right track, it looks like our differentials are incorrect exactly up to sign when differentiating wrt. something taken with an inverse.

Do you think it is a math mistake or an implementation mistake? 🤔

[mateuszbaran]

There isn't enough math comments left from the time I've added left-backward and right-forward actions to be sure. I'm just re-doing the math and checking it against implementation.

[mateuszbaran]

Could you check if apply_diff_group gives the correct result in this PR for group operation actions on SO(3) when a and p are not identity? I've made some mistakes in my math. The current state of implementation was derived from the Giles' paper I've cited but all these inverses and representation adjustments are easy to get wrong.

[mateuszbaran]

This fix was a little harder then I expected but I think this is finished now.