Group ring

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This page discusses the algebraic group ring of a discrete group; for the case of a topological group see group algebra, and for a general group see Group Hopf algebra.

In mathematics, a group ring is a ring R[G] constructed from a ring R and a group G (written multiplicatively). Sometimes the group ring is written simply as RG.

As an R-module, the ring R[G] is the free module over R on the elements G. If R is a field K, the group ring is called a group algebra; it is a vector space over K, with the basis vectors given by the elements of G. The elements of the group ring are finite linear combinations of elements of G with coefficients in R. Multiplication is defined by the group operation in G extended by linearity and distributivity, and the requirement that elements of R commute with elements of G. The identity element of G is the multiplicative identity of the ring R[G].

If R is commutative, then R[G] is an associative algebra over R, and it is also called the group algebra.

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[edit] Definition

Let G be a group and R a ring. We first define the set R[G] to be one of the following:

  • The set of all formal R-linear combinations of elements of G.
  • The free R-module with basis G.
  • The set of all functions f: G → R with f(g) = 0 for all but finitely many g in G.

No matter which definition is used, we can write elements of R[G] in the form \sum_{g \in G} a_g g, with all but finitely many of the ag being 0, and an addition is defined on R[G] (by addition of formal linear combinations, addition in the module, or addition of functions, respectively). Multiplication of elements of R[G] is defined by setting

(\sum_{g \in G} a_g g )( \sum_{h \in G} b_h h ) \ = \ \sum_{g,h \in G} (a_g b_h ) gh.

If R has a unit element, this is the unique bilinear multiplication for which (1 g)(1 h) = (1 gh). In this case, G is commonly identified with the elements 1 g of R[G]. The identity element of G then serves as the 1 in R[G].

R is commonly a commutative ring with unit, or even a field.

[edit] Two simple examples

Let G = Z3, the cyclic group of three elements with generator a. An element r of C[G] may be written as

r = z_1 + z_2 a + z_3 a^2\,

where z1, z2 and z3 are in C, the complex numbers. Writing a different element s as

s=w_1 +w_2 a +w_3 a^2\,

their sum is

r + s = z_1+w_1 + (z_2+w_2) a + (z_3+w_3) a^2\,

and their product is

rs = z_1w_1 + z_2w_3 + z_3w_2 +(z_1w_2 + z_2w_1 + z_3w_3)a +(z_1w_3 + z_3w_1 + z_2w_2)a^2\,

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.

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

[edit] Properties

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].

[edit] Group algebra over a finite group

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 over the field, with the elements being formal sums:

x=\sum_{g\in G} a_g g

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

g \cdot h = gh,

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

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:

e_g \cdot e_h = e_{gh}.

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

[edit] Regular 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.

Written as a representation, it is the representation g \mapsto ρg with the action given by \rho(g)\cdot e_h = e_{gh}, or

\rho(g)\cdot r = \sum_{h\in G} k_h \rho(g)\cdot e_h = \sum_{h\in G} k_h e_{gh}

[edit] Properties

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].

The group algebra C[G] of a finite group over the complex numbers is a semisimple ring. This result, Maschke's theorem, allows us to understand C[G] as a finite product of matrix rings with entries in C.

[edit] Representations of a group algebra

Taking K[G] to be an abstract algebra, one may ask for concrete representations of the algebra over a vector space V. Such a representation

\tilde{\rho}:K[G]\rightarrow \mbox{End} (V).

is an algebra homomorphism from the group algebra to the set of endomorphisms on V. Taking V to be an abelian group, with group addition given by vector addition, such a representation in fact a left K[G]-module over the abelian group V. That this is so is exhibited below, where each axiom of a module is demonstrated.

Pick r ∈ K[G] so that

\tilde{\rho}(r) \in \mbox{End}(V).

Then \tilde{\rho}(r) is a homomorphism of abelian groups, in that

\tilde{\rho}(r) \cdot (v_1 +v_2) = \tilde{\rho}(r) \cdot v_1 + \tilde{\rho}(r) \cdot v_2

for any v1, v2 ∈ V. Next, one notes that the set of endomorphisms of an abelian group is an endomorphism ring. The representation \tilde{\rho} is a ring homomorphism, in that one has

\tilde{\rho}(r+s)\cdot v = \tilde{\rho}(r)\cdot v + \tilde{\rho}(s)\cdot v

for any two r, s ∈ K[G] and v ∈ V. Similarly, under multiplication,

\tilde{\rho}(rs)\cdot v = \tilde{\rho}(r)\cdot \tilde{\rho}(s)\cdot v

Finally, one has that the unit is mapped to the identity:

\tilde{\rho}(1)\cdot v = v

where 1 is the multiplicative unit of K[G]; that is,

1 = e_e\,

is the vector corresponding to the identity element e in G.

The last three equations show that \tilde{\rho} is a ring homomorphism from K[G] taken as a group ring, to the endomorphism ring. The first identity showed that individual elements are group homomorphisms. Thus, a representation \tilde{\rho} is a left K[G]-module over the abelian group V.

Note that given a general K[G]-module, a vector-space structure is induced on V, in that one has an additional axiom

\tilde{\rho}(ar) \cdot v_1 + \tilde{\rho}(br) \cdot v_2 = a \tilde{\rho}(r) \cdot v_1 + b \tilde{\rho}(r) \cdot v_2 = \tilde{\rho}(r) \cdot (av_1 +bv_2)

for scalar a, b ∈ K.

Any group representation

\rho:G\rightarrow \mbox{Aut}(V),

with V a vector space over the field K, can be extended linearly to an algebra representation

\tilde{\rho}:K[G]\rightarrow \mbox{End}(V),

simply by mapping \rho(g) \mapsto \tilde{\rho}(e_g). Thus, representations of the group correspond exactly to representations of the algebra, and so, in a certain sense, talking about the one is the same as talking about the other.

[edit] Center of a group algebra

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

Z(K[G]) := \left\{ z \in K[G] \mid zr = rz \mbox{ for all } r \in K[G]\right\}.

The center is equal to the set of class functions, that is the set of elements that are constant on each conjugacy class: Z(K[G]) = \left\{ \sum_{g \in G} a_g g: \text{for all } g,h \in G, a_g = a_{h^{-1}gh}\right\}

If K=\mathbb{C}, the set of irreducible characters of G forms an orthonormal basis of Z(K[G]) with respect to the inner product \langle \sum_{g \in G} a_g g, \sum_{g \in G} b_g g \rangle = \frac{1}{|G|} \sum_{g \in G} \bar{a_g} b_g.

[edit] Group rings over an infinite group

Much less is known in the case where G is countably infinite, or uncountable, and this is an area of active research. 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.

If G is torsion-free, it is conjectured that C[G] has no nontrivial idempotents or zero divisors; this has been proven for special cases, such as the ones where G is abelian, elementary amenable, or free.

The case of G being a topological group is discussed in greater detail in the article on group algebras.

[edit] Representations of a group ring

A module M over R[G] is then the same as a linear representation of G over the field R. There is no particular reason to limit R to be a field here. However, the classical results were obtained first when R is the complex number field and G is a finite group, so this case deserves close attention. It was shown that R[G] is a semisimple ring, under those conditions, with profound implications for the representations of finite groups. More generally, whenever the characteristic of the field R does not divide the order of the finite group G, then R[G] is semisimple (Maschke's theorem).

When G is a finite abelian group, the group ring is commutative, and its structure is easy to express in terms of roots of unity. When R 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 Jacobson radical, and this gives the corresponding subject of modular representation theory its own, deeper character.

[edit] Category theory

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

\operatorname{GrpRng}\colon \mathbf{\operatorname{Grp}} \to R\mathbf{\operatorname{-Alg}}
\operatorname{GrpUnits}\colon R\mathbf{\operatorname{-Alg}} \to \mathbf{\operatorname{Grp}}

where "GrpRng" takes a group to its group ring over R, and "GrpUnits" takes an R-algebra to its group of units.

When R = Z, this gives an adjunction between the category of groups and the category of rings.

[edit] Generalizations

The group algebra generalizes to the categorical algebra, of which another example is the incidence algebra.

[edit] References