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Group ring

In mathematical representation theory, the group ring is an abstract algebra construction, that allows representations of a group G, over a field K, to be treated as modules.

The ring constructed (notation K[G], or sometimes just KG) can be described as the vector space over K with basis the elements g of G, and ring multiplication the group operation in G extended by bilinearity to the whole space. That is, g1g2 = g3 as an equation in G still holds true in K[G], and the whole structure of K[G] as an algebra over K follows when we apply the distributive law and K-linearity.

It is then true that a module M for K[G] is just what is normally meant as a linear representation of G over the field K. There is no particular reason to limit K to be a field here; but the classical results that were obtained first when K is the complex number field and G a finite group justify close attention to this case. It was shown that K[G] is a semisimple ring, under those conditions, with profound implications for the representations of finite groups.

When G is a finite abelian group, the group ring is commutative, and its structuer easy to express in terms of roots of unity. When K is a field of characteristic p, and the prime number p divides the order of the finite group G, then the group ring is not semisimple: it has a non-zero radical, and this gives the corresponding subject of modular representation theory its own, deeper character.

An example of a group ring of an infinite group is the ring of Laurent polynomials: this is exactly the group ring of an infinite cyclic group.

There is a neat characterisation from category theory of the group ring construction as adjoint to the functor taking a K-algebra to its group of units.

Continuous case

For the purposes of functional analysis, and in particular of harmonic analysis, one wishes to carry over the group ring construction to topological groups G. In this case the group ring should consist of enough functions on G, with product the convolution: in the algebraic case one already sees that if group ring elements are written as functions F(g) of group elements, with values in K, then the ring product is a type of convolution.

In practice there are various candidates for the complex group algebra of G. One can take the algebra of continuous complex functions on G that are zero outside a compact set: then the convolution product will present no convergence difficulties. In case G is compact this is just C(G), the ring of all continuous functions on G with the convolution product. In this way one will get a C-star algebra.

In general the need to apply Fubini's theorem leads to the use of L1(G), L2(G), and their intersection. That is, to deal with the analytical difficulties one needs a range of algebras, instead of just one.