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Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?

Assume that you don't know the irreducibles or the branching rules of the group. I am looking for a down-to-earth linear algebraic way to find the similarity transformation which will do the decomposition with no knowledge of the group's representation theory.

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    $\begingroup$ See math.stackexchange.com/a/185001/448 . If someone reading this knows a better place to point people than my answers, please do so! $\endgroup$
    – David E Speyer
    Commented Mar 13, 2015 at 21:32
  • $\begingroup$ @DavidSpeyer Thanks! Isn't my case simpler than what that other question asks for? I am saying - lets say you have $8$ $4\times 4$ matrices given to you explicitly. Then what would you do? $\endgroup$
    – user6818
    Commented Mar 13, 2015 at 21:44
  • $\begingroup$ Wait, is this a group in characteristic zero? Then I would compute the center of this group algebra (linear algebra) and diagonalize the central elements (since they commute). $\endgroup$
    – David E Speyer
    Commented Mar 13, 2015 at 23:49
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    $\begingroup$ Some old but rather explicit physics notes of mine on doing this in the context of a simple lattice gauge theory are here: drive.google.com/file/d/0ByTBBePgIzD0a3ZGSWt3aFlJS00/… $\endgroup$
    – Steve Huntsman
    Commented Mar 14, 2015 at 0:07
  • $\begingroup$ @DavidSpeyer I am working over complex numbers. So you say that I can just detect the centre of the group and diagonalize any one of them and this basis should simultaneously block-diagonalize the rest of the matrices? But finding the center is still a brute-force calculation - right? $\endgroup$
    – user6818
    Commented Mar 14, 2015 at 20:23

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