check_mul_axes specialization

[lkdvos]

This PR fixes #61, since the actual issue isn't flux conservation, the multiplied matrices are just not compatible, and the multiplication should error before even computing the result.

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[lkdvos]

The polar decompositions are failing because the current implementation is fundamentally incompatible with non-zero flux: we have no guarantee that U, S, Vh = tsvd(A) yields U and Vh that can be multiplied. (This was previously also the case as the tests are marked broken, but now the call already errors) I'd suggest to for now comment out these tests, merge this, and I'll fix this in #62 ?

Edit: the additional contraction failures are unrelated to this PR, and main also doesn't pass the tests right now: https://github.com/ITensor/GradedArrays.jl/actions/runs/17896032720

[mtfishman]

Thanks for investigating this. I didn't notice that in the example in #61 the sectors don't match, I agree that case should error earlier based on check_mul_axes.

I also see how polar decomposition of equivariant tensors is nontrivial to define since you need S to be properly square in order to form the isometry U * V. In the current implementation, is the issue that for equivariant tensors we make S block diagonal, which means in general it won't be square (i.e. the sectors in the codomain and domain of S might be permuted from each other)? Would a solution be that for the sake of the polar decomposition we gauge transform S so that it is square (or in general for thin SVD, make S square even if that means that sometimes it isn't block diagonal)?

I'm fine with merging this PR as it is and investigating these other issues in followup PRs. That's too bad about the test failures, I wonder how that slipped through the downstream testing... Maybe that was caused by the refactor of the contraction code in ITensor/TensorAlgebra.jl#75.

[lkdvos]

I indeed think we can fix the polar decomposition by making the right choices of the gauges, in particular not putting the flux on the S should be enough I think. I am tackling that in the other PR, I'll try and push my update once I'm past border control ;)

[mtfishman]

I'm fine with merging this PR as it is and investigating these other issues in followup PRs. That's too bad about the test failures, I wonder how that slipped through the downstream testing... Maybe that was caused by the refactor of the contraction code in ITensor/TensorAlgebra.jl#75.

I see, it appears the GradedArrays.jl isn't a downstream test of TensorAlgebra.jl, I'll fix that.