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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. 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.