I disagree. In general, you need a multilinear form of degree k where 1⊂(πλ)⊗k, while the number of mirror spaces is the degree of Im(πλ), You are very lucky that the rotations and mirroring you use come from the alternating representation of Sn where these happen to match, since sgn⊗sgn≅1 and Im(sgn)=±1.
I am in the process of writing a much longer comment, but I think the primary question your post leaves unanswered is, “why a bilinear form, not any other degree?” and pulling on that thread unravels the understanding this post allegedly gives.
The bilinear form is from the inner product. The inner product is generally defined as ‘antilinear in first argument, linear in second’. If replacing the first argument with complex conjugate space, it is now ‘linear in first argument, linear in second argument’, i.e. bilinear. This is of course for a single Hilbert space (and its complex conjugate). When having multiple Hilbert spaces tensored together, the ‘tensor product of Hilbert spaces’ yields an inner product for that tensored space. That is implicitly multilinear, although decomposes as usual for tensor product, into multiple bilinearities.
I disagree. In general, you need a multilinear form of degree k where 1⊂(πλ)⊗k, while the number of mirror spaces is the degree of Im(πλ), You are very lucky that the rotations and mirroring you use come from the alternating representation of Sn where these happen to match, since sgn⊗sgn≅1 and Im(sgn)=±1.
I am in the process of writing a much longer comment, but I think the primary question your post leaves unanswered is, “why a bilinear form, not any other degree?” and pulling on that thread unravels the understanding this post allegedly gives.
The bilinear form is from the inner product. The inner product is generally defined as ‘antilinear in first argument, linear in second’. If replacing the first argument with complex conjugate space, it is now ‘linear in first argument, linear in second argument’, i.e. bilinear. This is of course for a single Hilbert space (and its complex conjugate). When having multiple Hilbert spaces tensored together, the ‘tensor product of Hilbert spaces’ yields an inner product for that tensored space. That is implicitly multilinear, although decomposes as usual for tensor product, into multiple bilinearities.