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.
G1. Group Representations and Examples
Definition. A representation of a group G is 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.
- 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.
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 S3, dim(V) = 3! = 6.
G2. The Group Algebra
Definition. Given field K and finite group G, the group algebra K[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:
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:
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 V1 and V2, 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 HomK(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 HomK(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) HomK(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.