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I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere else. Are there problems in (or applications of) the theory of group rings that are separate from representation theory? If so, where could I read about them?

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Of course, one can study group rings as objects in their own right! One long-standing question is, if I recall correctly, whether every group rings has a non-trivial unit. Finite groups give you group rings which do, and Graham Higman introduced "Locally Indicable" groups as examples of groups which give you group rings with non-trivial units (a group is locally indicable if every proper, non-trivial subgroup maps onto the infinite cyclic group). –  user1729 Mar 17 '13 at 19:25

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It is hard to give a definitive answer to your question, because many branches of mathematics are related to representation theory or they have an interpretation in terms of representation theory. For example, module theory over a ring $R$ can be interpreted as the representation theory of $R$.

However, I can give an example for what you asked. In homological algebra, it is proven that the homology of a group $G$ is isomorphic to the Hochschild homology of $\mathbb{C}G$, the group algebra of $G$. A good reference for this statement is the Weibel's book: "An introduction to homologial algebra, Cambridge University Press, (1994)".

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I think you meant Weibel's "An Introduction to Homological Algebra"? –  A.P. Mar 17 '13 at 22:16

In addition to Vahid Shirbisheh's answer:

Error-correcting codes

Skew fields (Malcev-Neumann Theorem)

Banach algebras

and so on.

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