Group ring


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 one-to-one with 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 given ring is commutative, a group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring.

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

Contents

Definition

Let G be a group, written multiplicatively, and let R a ring. The group ring of G over R, which we will denote by R[G], is the set of mappings f : GR of finite support,[1] where the product αf of a scalar α in R and a vector (or mapping) f is defined as the vector x \mapsto \alpha \cdot f(x), and the sum of two vectors f and g is defined as the vector x \mapsto f(x) + g(x). To turn the commutative group R[G] into a ring, we define the product of f and g to be the vector

x\mapsto\sum_{uv=x}f(u)g(v).

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"[2]:

\sum_{g\in G}f(g) g,

or simply

\sum_{g\in G}f_g g.

where this doesn't cause confusion.[1]

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_0 + z_1 a + z_2 a^2\,

where z0, z1 and z2 are in C, the complex numbers. Writing a different element s as

s=w_0 +w_1 a +w_2 a^2\,

their sum is

r + s = (z_0+w_0) + (z_1+w_1) a + (z_2+w_2) a^2\,

and their product is

rs = (z_0w_0 + z_1w_2 + z_2w_1)
+(z_0w_1 + z_1w_0 + z_2w_2)a
+(z_0w_2 + z_2w_0 + z_1w_1)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 over a ring R: these are nothing more or less than the group ring of the infinite cyclic group Z over R.

Some basic properties

Assuming that the ring R has a unit element 1, and denoting the group unit by 1G, the ring R[G] contains a subring isomorphic to R, and its group of invertible elements contains a subgroup isomorphic to G. For considering the indicator function of {1G}, which is the vector f defined by

f(g)=
1\cdot 1_G + \sum_{g\not= 1_G}0 \cdot g=
\mathbf{1}_{\{1_G\}}=\begin{cases}
1\text{ if }g = 1_G \\
0\text{ if }g \ne 1_G
\end{cases},

the set of all scalar multiples of f is a subring of R[G] isomorphic to R. And if we map each element s of G to the indicator function of {s}, which is the vector f defined by

f(g)=
1\cdot s + \sum_{g\not= s}0 \cdot g=
\mathbf{1}_{\{s\}}=\begin{cases}
1\text{ if }g = s \\
0\text{ if }g \ne s
\end{cases}

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].

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}.

Interpretation as functions

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 function KG := Hom(G,K) are dual: given an element of the group algebra x = \sum_{g\in G} a_g g and a function on the group f\colon G \to K, these pair to give an element of K via (x,f) = \sum_{g\in G} a_g f(g), which is a well-defined sum because it is finite.

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 gρg 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}.

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.

Group rings satisfy a universal property.[1]

Every group ring of a finite group over a field is a Frobenius algebra, with Frobenius form σ(a,b) the coefficient of the identity element of a·b.

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 rK[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, v2V. 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, sK[G] and vV. 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, bK.

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.

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.

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.

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:

  • Unique product groups (which include virtually abelian groups, orderable groups, and free groups, since they are orderable)
  • Diffuse groups - in particular, groups that act freely isometrically on R-trees, and the fundamental groups of surface groups except for the fundamental groups of direct sums of one, two or three copies of the projective plane.

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

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.

Category theory

Adjoint

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, and the unit of the adjunction takes a group G to a group that contains trivial units: G \times \{\pm 1\}=\{\pm g\}. In general, group rings contain nontrivial units. If G contains elements a and b such that an = 1 and b does not normalize \langle a\rangle then the square of x = (a − 1)b(1 + a + a2 + ... + an − 1) is zero, hence (1 + x)(1 − x) = 1. The element 1 + x is a unit of infinite order.

Generalizations

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

Filtration

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.

See also

Representation theory

Category theory

Notes

  1. ^ a b c Polcino & Sehgal (2002), p. 131.
  2. ^ Polcino & Sehgal (2002), p. 129 and 131.

References


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