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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Samuel Handwich
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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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In addition to Vahid Shirbisheh's answer:
Error-correcting codes
Skew fields (Malcev-Neumann Theorem)
Banach algebras
and so on.
Boris Novikov
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Yeah group rings has a lot of stuff. It is an integral part of Ring theory and has provided many good counterexamples being mostly non commutative rings. You can study them completely without representation theory as an independent subject. sehgal and passman, sehgal 2 sehgal 3 are some references
Bhaskar Vashishth
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