In what sense is $SU(n)$ a finite group?  Perhaps your question is about compact (including finite) groups?  In that case, the representation theory is very well understood.  For compact groups, my favourite reference is the book by Bröcker and tom Dieck [Represetations of compact Lie groups][1].  Section II.7 describes the representation ring in general and introduces the Adams operations,... and then Section VI.5 contains the structure of the representation rings for the simply-connected classical Lie groups.


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I should have added that the representations only form a semiring under direct sum and tensor product.  The representation ring is obtained from this by the standard Grothendieck construction.  You essentially have to add so-called *virtual representations*.

The book I mentioned does not treat the case of infinite-dimensional representations, though.  Perhaps someone here can say more about this case.


  [1]: http://books.google.com/books?id=AfBzWL5bIIQC