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?
What is the difference between direct product and direct sum of a finite number of group representations.
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vector-spaces
representation-theory
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0Check for example page 10 and 11 – 2012-07-01
1 Answers
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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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1I'd like to add a link to a lucid presentation aiming to clear up the different notions of *direct sum*, *direct product* and the *tensor direct product* [http://suchideas.com/articles/maths/math-phys/tensor-direct-products-vs-direct-sums/](http://suchideas.com/articles/maths/math-phys/tensor-direct-products-vs-direct-sums/) – 2014-01-16