Character theory is one of the most beautiful topics in undergraduate mathematics; the objective is to study the structure of a finite group *G* by letting it act on vector spaces. Earlier, we had already seen some interesting results (e.g. proof of the Sylow theorems) by letting *G* act on finite sets. Since linear algebra has much more structure, one might expect an even deeper theory.

Some prerequisites for understanding this set of notes:

- basic group theory, up to group quotients and homomorphisms;
- linear algebra, including tensor product of vector spaces;
- elementary theory of (left) modules over non-commutative rings, including up to module quotients and homomorphisms.

[ We’ve yet to cover module theory and linear algebra; hopefully this will be rectified in the future. ]

At one point, one also needs to take the tensor product , where *M* is an *R*-module and are (non-commutative) rings. But this is a rather minor aspect, and we’ll also describe the explicit construction so the reader can just accept some of the results at face value for now.

Throughout this document, *G* denotes a finite group and all linear algebra is performed over a field *K*. As time passes by, we’ll restrict ourselves to fields of characteristic 0, and then finally to the complex field **C **(or any of your favourite algebraically closed fields of characteristic 0). Also, all vector spaces over *K* are assumed to be of finite dimension.

Let’s begin.

## G1. Group Representations and Examples

We define:

Definition. Arepresentationof a groupGis a group homomorphism where V is a finite-dimensional vector space over field K.

If we fix a basis for *V*, then this is tantamount to giving a group homomorphism , where *n* = dim(*V*). Thus, each element of *G* now corresponds to an *n* × *n* matrix with entries in *K* such that product in *G* corresponds to product of matrices.

[ **Note**: throughout all notes on this site, matrix representation for a linear map is obtained via , where *M* is a matrix and *v* is a column vector. Thus, if dim(*V*)=*m*, dim(*W*)=*n* and *T* : *V* → *W*, then the underlying matrix has *m* columns and *n* rows, i.e. *n* × *m*. ]

Just like the case of group actions, we can think of a group representation as providing a map:

which is conveniently denoted *g**·**v* instead. This satisfies and for all group elements and Under this notation, one also says *G* **acts on** *V*.

**Examples**

- Let dim(
*V*)=1 and*G*act trivially on it. Thus takes every*g*to 1. We call this the**trivial representation**. - Suppose is the full symmetric group. Let dim(
*V*)=1 and let*G*act on it via where sgn(*g*) = +1 if*g*is an even permutation and -1 if it’s odd. This is called the**alternating representation**. Note that it’s only available for and not for any old group. - Let again, and dim(
*V*)=*n*be spanned by the basis Now acts on*V*by taking E.g. if*n*= 3, the representation is:

- Let be a cyclic group of order 3 and dim(
*V*)=2. A representation of*G*is given by: Since this matrix is of order 3, the map is well-defined.

**Regular Representation**

Example 3 above is clearly generalisable: if *G* acts on finite set *X*, then let *V* be a vector space with abstract basis Thus, dim(*V*) = #*X*. Now acts on *V* by taking

In particular, any group *G* acts on itself by left multiplication, so this gives a representation of dimension #*G*. Explicitly, *V* is given an abstract basis and the action of is given by:

This is called the **regular representation** of group *G*. Note that example 3 is *not* the regular representation since in the regular representation of *S*_{3}, dim(*V*) = 3! = 6.

## G2. The Group Algebra

We define:

Definition. Given field K and finite group G, thegroup algebraK[G] is a K-vector space with an abstract basis given by:

and multiplication given by and extended linearly.

Some concrete computations will make it much clearer. Suppose and *K*=**C**. Then a typical product of elements of **C**[*G*] looks like:

The following should now be clear.

Theorem. The group algebra K[G] is a ring which contains K as a subring. It is commutative if and only if G is abelian.

As a ring, we can talk about left modules over *K*[*G*]. These turn out to correspond precisely to representations of *G*.

Let’s do the easy direction first: suppose we’re given a left *K*[*G*]-module *V*. Then *V* is naturally a *K*-vector space and we obtain an action of *G* on *V* by restricting the left-module action to the basis Since for any we get a representation of *G* on *V*.

Conversely, suppose *G* acts on *V* via *K*-linear maps, i.e. every gives rise to a linear map We’ll define a *K*[*G*]-module structure on *V*, by first decreeing that act on *V* via ρ(*g*), then extending linearly to the whole *K*[*G*]. Explicitly:

takes

**Concrete Example**

Consider example 4 from section G1, where is cyclic of order 3 and the representation takes *a* to Now a typical element of *K*[*G*] is of the form:

where

The corresponding matrix is then:

or . This represents the action of on *V* as a *K*[*G*]-module.

## G3. Creating New Representations

We’ll look at ways to create new representations of *G* from existing ones.

**A. Direct Sum**

If *R* is a ring, then the direct sum of two *R*-modules is another one. In particular, this holds for *R* = *K*[*G*] as well. Specifically, if and are both representations, then the direct sum gives:

If we pick bases of *V*_{1} and *V*_{2}, then the resulting basis of gives the matrix of *g* : *V* → *V* as

**B. Submodules and Quotients**

Generally, if *M* is a left *R*-module and a submodule, we get a quotient module *M*/*N*. When *V* is a left *K*[*G*]-module, a submodule is said to be a **G****-invariant subspace**. Clearly, this is a vector subspace; also, for each the action of *g* on *V* results in Conversely, if *W* is a vector subspace of *V* which is invariant under all then it is a *K*[*G*]-submodule.

If we pick a basis of *W* and extend it to *V*, then the matrix representation of is:

If *V* and *W* are *K*-vector spaces, we can take their tensor product over *K*: . Explicitly, if is a basis of *V* and a basis of *W*, then gives a basis of the tensor product *X*.

Given , since the action is linear on both *V* and *W*, this induces a linear map

Note that ; indeed, on elements this is easily seen to be true:

Since the set of all such elements spans the result follows. In terms of matrix representation, we get:

**D. Space of Linear Functions**

Suppose *V* and *W* are *K*[*G*]-modules. The space of all *K*-linear maps is also a *K*[*G*]-module. To define the action of *G* on *X*, let’s imagine a *K*-linear map *f* : *V* → *W* written in the form of a huge lookup table (*v*, *f*(*v*)) such that each *v* occurs exactly once on the left. Now let *G* act on the entire table by replacing (*v*, *f*(*v*)) with the pair (*g**·**v*, *g*·*f*(*v*)). Unwinding the definition, we see that *G* acts on *X* via:

Note that in the composition *gfg*^{-1}, the left *g* acts on *W* while the right *g*^{-1} acts on *V*.

**E. Dual Space**.

A special case of the above is when *W* = *K* with the trivial representation. The resulting Hom_{K}(*V*, *K*) is known in linear algebra as the **dual space** *V**. The above definition then gives us an action of *G* on *V** via:

Let’s do some sanity check here. From linear algebra, there’s a canonical isomorphism:

If both *V* and *W* are *K*[*G*]-modules, then there appears to be two different ways to define a *G*-action on Hom_{K}(*V*, *K*). Fortunately, both ways are identical; this can be checked by letting act on the element on the left and the map on the right.

- On the left, we get .
- On the right, we get the composition which is the image of .

## In a Nutshell

Given a finite group *G* and field *K*, we’ve defined the group algebra *K*[*G*] which is a ring containing *K*. This is done by using an abstract basis so that the dimension of *K*[*G*] is precisely the order of *G*. Product is defined via and extended linearly.

There’s a one-to-one correspondence between (1) *K*[*G*]-modules, and (2) linear representations of *G* on *K*-vector spaces.

The usual operations to construct new *K*[*G*]-modules are (A) direct sums, (B) submodules and quotients, (C) tensor products, (D) Hom_{K}(*V*, *W*) and (E) duals.

Everything presented so far is rather generic; in fact, one could even take *K* as any commutative ring and there’d be no effect on the theory thus far. In the next installation, we’ll explore the structure of *K*[*G*]-modules in greater detail.