In the representation theory, the representation ring is defined and some results can be expressed with the representation ring R(G). What is the benefit of having this extra definition and what insights can it provide?
What is the use of representation ring
1 Answers
It is often useful, when one has many objects of a kind —in your case, representations of a group— it is useful to collect them in some efficient package —in your case, the representation ring— which, even if it loses some of the information, retains other parts.
The representation ring allows you to solve many questions about representations of a group without actually knowing them. In a way, it is a more abstract version of the character table, and you probably know that one can often compute the character table of a group without knowing the representions themselves. This is also true of the $R(G)$.
The way to answer your question is to read about what people actually do with $R(G)$. Serre's book on representations of finite groups has a delightful exposition of the theory. Likewise, the book by Fulton and Harris has some applications of the representation ring in the context of Lie groups---and generally, applications are all over the place.