Group ring

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra.

The apparatus of group rings is especially useful in the theory of group representations.

The summation is legitimate because f and g are of finite support, and the ring axioms are readily verified.

Some variations in the notation and terminology are in use. In particular, the mappings such as f : GR are sometimes written as what are called "formal linear combinations of elements of G, with coefficients in R":[1]

Note that if the ring R is in fact a field K, then the module structure of the group ring RG is in fact a vector space over K.

Notice that the identity element 1G of G induces a canonical embedding of the coefficient ring (in this case C) into C[G]; however strictly speaking the multiplicative identity element of C[G] is 1⋅1G where the first 1 comes from C and the second from G. The additive identity element is zero.

When G is a non-commutative group, one must be careful to preserve the order of the group elements (and not accidentally commute them) when multiplying the terms.

2. A different example is that of the Laurent polynomials over a ring R: these are nothing more or less than the group ring of the infinite cyclic group Z over R.

Multiplication, as in any other group ring, is defined based on the group operation. For example,

the resulting mapping is an injective group homomorphism (with respect to multiplication, not addition, in R[G]).

If R and G are both commutative (i.e., R is commutative and G is an abelian group), R[G] is commutative.

If H is a subgroup of G, then R[H] is a subring of R[G]. Similarly, if S is a subring of R, S[G] is a subring of R[G].

If G is a finite group of order greater than 1, then R[G] always has zero divisors. For example, consider an element g of G of order |g| = m > 1. Then 1 - g is a zero divisor:

For example, consider the group ring Z[S3] and the element of order 3 g=(123). In this case,

Group algebras occur naturally in the theory of group representations of finite groups. The group algebra K[G] over a field K is essentially the group ring, with the field K taking the place of the ring. As a set and vector space, it is the free vector space on G over the field K. That is, for x in K[G],

The algebra structure on the vector space is defined using the multiplication in the group:

where on the left, g and h indicate elements of the group algebra, while the multiplication on the right is the group operation (denoted by juxtaposition).

Because the above multiplication can be confusing, one can also write the basis vectors of K[G] as eg (instead of g), in which case the multiplication is written as:

Thinking of the free vector space as K-valued functions on G, the algebra multiplication is convolution of functions.

While the group algebra of a finite group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of finite sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology), these correspond to functions with compact support.

However, the group algebra K[G] and the space of functions KG := Hom(G, K) are dual: given an element of the group algebra

and a function on the group f : GK these pair to give an element of K via

Taking K[G] to be an abstract algebra, one may ask for representations of the algebra acting on a K-vector space V of dimension d. Such a representation

The group algebra is an algebra over itself; under the correspondence of representations over R and R[G] modules, it is the regular representation of the group.

The dimension of the vector space K[G] is just equal to the number of elements in the group. The field K is commonly taken to be the complex numbers C or the reals R, so that one discusses the group algebras C[G] or R[G].

where dk is the dimension of Vk. The subalgebra of C[G] corresponding to End(Vk) is the two-sided ideal generated by the idempotent

For a more general field K, whenever the characteristic of K does not divide the order of the group G, then K[G] is semisimple. When G is a finite abelian group, the group ring K[G] is commutative, and its structure is easy to express in terms of roots of unity.

When K is a field of characteristic p which divides the order of G, the group ring is not semisimple: it has a non-zero Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character.

The center of the group algebra is the set of elements that commute with all elements of the group algebra:

The center is equal to the set of class functions, that is the set of elements that are constant on each conjugacy class

If K = C, the set of irreducible characters of G forms an orthonormal basis of Z(K[G]) with respect to the inner product

Much less is known in the case where G is countably infinite, or uncountable, and this is an area of active research.[3] The case where R is the field of complex numbers is probably the one best studied. In this case, Irving Kaplansky proved that if a and b are elements of C[G] with ab = 1, then ba = 1. Whether this is true if R is a field of positive characteristic remains unknown.

A long-standing conjecture of Kaplansky (~1940) says that if G is a torsion-free group, and K is a field, then the group ring K[G] has no non-trivial zero divisors. This conjecture is equivalent to K[G] having no non-trivial nilpotents under the same hypotheses for K and G.

In fact, the condition that K is a field can be relaxed to any ring that can be embedded into an integral domain.

The conjecture remains open in full generality, however some special cases of torsion-free groups have been shown to satisfy the zero divisor conjecture. These include:

The case where G is a topological group is discussed in greater detail in the article Group algebra of a locally compact group.

Categorically, the group ring construction is left adjoint to "group of units"; the following functors are an adjoint pair:

Any other ring satisfying this property is canonically isomorphic to the group ring.

The group algebra generalizes to the monoid ring and thence to the category algebra, of which another example is the incidence algebra.

If a group has a length function – for example, if there is a choice of generators and one takes the word metric, as in Coxeter groups – then the group ring becomes a filtered algebra.