Today, I want to talk a bit about group representation theory. Many of us (such as myself) are interested in representation theory in general and will likely talk more about it in the future, so it will be useful to summarize the essential ideas here to refer back. But the basics are well known and have been discussed at length on other blogs (see, e.g. here, which is discussing the subject right now), so I am merely going to summarize these facts without proofs. The interested reader can read these notes for full details. Then, I’ll mention a property to be used later on.
What is a group representation?
Start with a group . At least for now, we’re essentially going to be constructed with finite groups, but many of these constructions generalize. A representation of is essentially an action of on a finite-dimensional complex vector space .
Formally, we write:
Definition 1 A representation of the group is a finite-dimensional complex vector space and a group-homomorphism . In other words, it is a group homomorphism , where , and is the group of invertible -by- matrices.
An easy example is just the unit representation, sending each to the identity matrix.
Alternatively, we can form the group ring of ; this consists of formal sums for coefficients . It is made into a ring by pointwise multiplication: we demand that multiplication in the group ring match multiplication in the group, and be linear.
Phrased in this language, a representation of is just a -module.
One of the nice facts about group representation theory is that we can split a representation of a group into “atoms” which contain no smaller elements. Formally, we write:
Definition 2 A nonzero representation of the group is said to be simple or irreducible if (which is a module) contains no proper -modules other than zero.
The following is Maschke’s theorem, which we state without proof:
Theorem 3 A representation of can be written as a direct sum,
where the ‘s are simple representations, and the are integers.
The decomposition is also unique up to isomorphism. (Note that the direct sum of representations is a representation itself.)
This is equivalent to saying that the group ring is semisimple.
[Edit, 6/14- for a proof of Maschke’s theorem, see this post, which appeared after this one..]
The Regular Representation
Any ring is automatically a (left) -module. In particular, the group ring is itself a -module, which is also a finite-dimensional vector space, if is a finite group.
Thus the underlying vector space of is a representation of . This is called the regular representation.
The following is the fundamental property of the regular representation:
Theorem 4 When decomposed into simple components, the regular representation of a finite group contains each simple component with multiplicity equal to the dimension of that component.
So, in particular, there are only finitely many such (non-isomorphic) simple representations, a fact which can be proved more generally for semisimple algebras.
Characters and Class Functions
Given a representation for a vector space, we define the corresponding function by .
Definition 5 is called the character of .
From the basic properties of traces, characters are additive on direct sums, i.e. . Moreover, characters are multiplicative on tensor products.
From the properties of traces (specifically for matrices ), we see that the characters are class functions; that is, for all . The characters are invariant under inner automorphisms.
Theorem 6 The irreducible characters of form a basis of the vector space of class functions on .
In particular, this implies the number of irreducible representations of , up to isomorphism, is given by the number of conjugacy classes.
This basis is actually orthonormal with respect to the Hermitian inner product on the space of class functions defined as follows:
because of the following:
Theorem 7 For irreducible representations , we have if are isomorphic, and otherwise.
In particular, putting Maschke’s theorem and this together, we see that to find the multiplicties with which an irreducible occurs in , one just takes the inner product .
I want to end this post with a random remark that will become useful later on.
First of all, our matrices for defining a representation are actually diagonalizable. We know from linear algebra that we can put such a matrix in a triangular form, but we also know that some power of them is the identity: . So fix , a triangular matrix, which satisfies for some . I claim is diagonal. Indeed, the condition means , or satisfies . In particular, the minimal polynomial of divides . Since has no repeated roots, the minimal polynomial also has none. By a general theorem of linear algebra, this implies is diagonalizable. Moreover, let the diagonal entries be . Then each satisfies .
In this case, the trace is just the sum . It follows that the character is always a sum of -th roots of unity and is in particular an algebraic integer.
This fact will be useful to us later, as we talk about applications of group representation theory. It will also enable us to show that the dimensions of group representations divide the group order. For now though, I’ll leave it here.