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In what homological degree do you put $V$ when you define the "exterior algebra"? In all contexts where we deal with cohomology and homotopy, it is really much better to say "symmetric algebra on the degree shift of $V$" and specify the degree shift, to avoid ambiguity.
@CaveJohnson thanks! In principle it is not clear that the two sequences coincide (with each other or with the sequence I am thinking of). But it is an interesting angle. Alas that paper has nothing on joint rank, which is what my question is about.
If by any chance you will want to see the paper (even though you do not speak Russian), this is the entire collection of papers "Matrix Problems": dropbox.com/scl/fi/drk0uvgn2jykmx4cpbni5/…
Are you aware of the De Concini-Procesi construction (sort of the same as Kapranov but more general - in the context of "wonderful compactifications" and in a sense more explicitly traced in many places in the literature)?
@PaceNielsen I am very surprised to hear that in your area not a lot goes to the arXiv! My experience in hiring committees suggests that people do use arXiv a lot to have a first approximation of candidates' research activity, and so if some fields do not use arXiv that much it really should be better known...
@SamHopkins not only this depends quite drastically on the domain of maths, I also feel that the very drastic increase of the number of daily submissions over the last 20 years impacted the quality quite drastically too. I see more and more papers that benefited no proof-reading by the authors (and it is very easy to hide mistakes in badly written arguments)...
@RBega2 QR factorization happens over reals, so it is way too restrictive, I'd say, plus, diagonalizing a transposition is a much more trivial task in any case.
Oh this (1)-(2) definition is really cute! I haven't seen it anywhere before. But it is probably hard for beginners. Of course elementary operations are just multiplications by elementary matrices so it is sufficiently straightforward (though not easy for beginners) to show that this gives the usual determinant. (Handling elementary matrices is done by saying that matrices of transpositions are diagonalizable, and that elementary transformations that add a multiple of a row to another row can be written as commutators.)
