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I have been reading in Fulton & Harris's book on representation theory and it talks about things like the decomposition of a direct product of representations $ V \otimes V $ into a direct sum of representations. It seems to me there is a real difference between a finite direct sum of representations and a direct product even though they agree on vector spaces. Is this difference real and if so , what is it?

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  • $\begingroup$ Can you tell on what page it talks about direct product ? $\endgroup$
    – user10676
    Jul 1, 2012 at 15:52
  • $\begingroup$ Check for example page 10 and 11 $\endgroup$ Jul 1, 2012 at 15:58

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$\otimes$ denotes the tensor product, not the direct product. This is different even for vector spaces; the tensor product of vector spaces of dimensions $m, n$ has dimension $mn$ rather than $m+n$ for the direct product or direct sum.

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