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JoshuaZ comments on What are you working on? - Less Wrong Discussion
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I'm working on tightening the bounds on the Jordan-Schur theorem. I've improved the best known bounds but not by much. If this project does succeed it might end up turning into my PhD thesis.
Out of idle curiosity (I haven't studied any proof of the Jordan-Schur theorem), are you doing that by tuning up the existing proofs of best bounds through more careful analysis, by replacing relatively large chunks of those proofs by new arguments, or by employing an altogether new proof?
Right now, I'm focused on tightening a specific lemma that turns out to be of independent interest and is used in a few other contexts also.
For a given n x n matrix X with coefficients in C, let ||X|| denote the Frobenius norm. (The Frobenius norm is essentially just Euclidean distance where one treats each matrix coefficient as two Euclidean coordinates, one from its real part and one from its imaginary part.)
Lemma: Let A and B be unitary matrices, and let C be the commutator of A and B (that is, C=ABA^(-1)B^(-1)). Then || I - C ||^2 <= 2|| I -A ||^2 || I -B||^2.
This Lemma is a standard step in the proof and it turns out that the strength of this inequality is one of the major limiting issues on improving the bounds. So I've been working on tightening that lemma. I've been somewhat successful in improving the lemma with the same hypotheses (i.e. just that A and B are unitary), but it turns out that for purposes of Jordan-Schur, one can without any trouble assume that A and B themselves generate a finite group, so I'm trying to see if I can substantially improve things with a version of the lemma that uses that additional assumption.