7. Irreducible representations

We define a class of representations that will provide us with building blocks for all possible representations.

Definition (Irreducible representation)

A representation of a group on a vector space is irreducible if it has no proper non-zero subrepresentations

Examples (Irreducible representation)

• All one-dimensional representations are irreducible.

• The tautological representation \(T\) of \(D_4\) is irreducible over real numbers. If there was some proper, non-zero subrepresentation, it would have to be one-dimensional, but no line in the place is left invariant under the action of the symmetry group of the square.

Counterexample (Irreducible representation)

The vertex permutation representation of \(D_4\) is not irreducible since the line spanned by the vector \((1,1,1,1)\) is a non-zero proper subrepresentation.

7.1. Complete reducibility

One of the most important statements in this lecture concerns reducibility of finite representations of finite groups.

Proposition

Every finite-dimensional representation of a finite group over the real or complex numbers decomposes into a direct sum of irreducible subrepresentations.

The key statement for the above proposition is that every subrepresentation of a real or complex representation of a finite group has a representation complement. That is, if \(W\) is a subrepresentation of \(V\), then there exists another subrepresentation \(W'\) of \(V\) such that \(V\cong W\oplus W'\) as representations of \(G\).

We will not prove this proposition in full generality, but instead we provide an algorithm on how to find the representation complement:

I. Let \(W\) be a subrepresentation of \(G\) inside a representation \(V\).

As an example, let us take \(V=\mathbb{R}^4\) to be the vertex permutation representation of \(D_4\), and \(W\) to be the subrepresentation spanned by \((-1,1,-1,1)\).

II. Fix any vector space complement \(U\) of \(W\) inside \(V\), and decompose \(V\) as the direct sum of vector spaces \(V\cong W\oplus U\).

Let us take \(U\) to be the subspace in \(\mathbb{R^4}\) consisting of all vectors whose last coordinate is zero. Then any element \((x_1,x_2,x_3,x_4)\in \mathbb{R}^4\) can be decomposed as

\[ (x_1,x_2,x_3,x_4)=(-x_4,x_4,-x_4,x_4)+(x_1+x_4,x_2-x_4,x_3+x_4,0)\in W\oplus U \]

III. The decomposition allows us to define a projection \(\Pi:V\to W\) onto the first factor: for all \(v\in V\) we can write \(v=w+u\) for some \(w\in W\) and \(u\in U\). Then \(\pi(v)=w\).

In our case

\[ \Pi((x_1,x_2,x_3,x_4))=(-x_4,x_4,-x_4,x_4) \]

IV. Define a new linear map \(\phi:V\to W\) obtained by averaging \(\Pi\) over all elements in \(G\):

\[ \phi(v)=\frac{1}{|G|}\sum_{g\in G}g.\Pi(g^{-1}.v) \]

\[\begin{split} \begin{array}{c|c|c|c|c|c|c|c|c|c} \hline g&e&(1234)&(13)(24)&(1432)&(13)&(24)&(12)(34)&(14)(23)\\ \hline g^{-1}.(x_1,x_2,x_3,x_4)&(x_1,x_2,x_3,x_4)&(x_2,x_3,x_4,x_1)&(x_3,x_4,x_1,x_2)&(x_4,x_1,x_2,x_3)&(x_3,x_2,x_1,x_4)&(x_1,x_4,x_3,x_2)&(x_2,x_1,x_4,x_3)&(x_4,x_3,x_2,x_1)\\ \hline \Pi(g^{-1}.x)&(-x_4,x_4,-x_4,x_4)&(-x_1,x_1,-x_1,x_1)&(-x_2,x_2,-x_2,x_2)&(-x_3,x_3,-x_3,x_3)&(-x_4,x_4,-x_4,x_4)&(-x_2,x_2,-x_2,x_2)&(-x_3,x_3,-x_3,x_3)&(-x_1,x_1,-x_1,x_1)\\ \hline g.\Pi(g^{-1}.x)&(-x_4,x_4,-x_4,x_4)&(x_1,-x_1,x_1,-x_1)&(-x_2,x_2,-x_2,x_2)&(x_3,-x_3,x_3,-x_3)&(-x_4,x_4,-x_4,x_4)&(-x_2,x_2,-x_2,x_2)&(x_3,-x_3,x_3,-x_3)&(x_1,-x_1,x_1,-x_1)\\ \hline \end{array} \end{split}\]

Summing over all 8 elements we get:

\[ \phi((x_1,x_2,x_3,x_4))=\frac{1}{8}(2x_1-2x_2+2x_3-2x_4)(1,-1,1,-1) \]

V. Find the kernel of \(\phi\) and set \(W'=\mathrm{ker}\phi\). Then \(W'\) is the representation complement we look for, and \(V\cong W\oplus W'\)

The kernel of \(\phi\) consists of all vectors that are mapped to \(0\). Therefore

\[ \mathrm{ker}\phi=\{(x_1,x_2,x_3,x_4):x_1-x_2+x_3-x_4=0\}=W' \]

Then \(W'\) is the representation complement of \(W\).

This algorithm allows is to find a complete decomposition for any finite dimensional representation \(V\) by following these three steps:

1. Find an invariant subspace \(W\) of \(V\)

2. Find the representation complement \(W\)’of \(W\) in \(V\)

3. Repeat steps 1. and 2. for \(W\) and \(W'\) until you end up with irreducible representations.

The decomposition into irreducible subrepresentations is not uniques in general. It is however unique up to an isomorphism in the following way:

Proposition

Let \(V\) be a finite-dimensional real or complex representation of a finite group \(G\). Then \(V\) has a decomposition into subrepresentations

\[ V=V_1\oplus V_2\oplus\ldots\oplus V_t \]

where each \(V_i\) is isomorphic to a direct sum of some number of copies of some fixed irreducible representation \(W_i\), with \(W_i\neq W_j\) unles \(i=j\). That is, given two different decompositions of \(V\) into non-isomorphic irreducible subrepresentations

\[ W_1^{a_1}\oplus \ldots\oplus W_t^{a_t}=U_1^{b_1}\oplus \ldots\oplus U_r^{b_r} \]

where the \(W_i\) (respectively \(U_i\)) are all irreducible and non-isomorphic, then after relabelling \(t=r\), \(a_i=b_i\), the subrepresentations \(W_i^{a_u}\) equal the \(U_i^{b_i}\) for all \(i\), and the corresponding \(W_i\) are isomorphic to \(U_i\) for all \(i\).

7.2. Classification of irreducible representations of finite groups

In this part of the module we want to find all irreducible representaions for finite groups, with special emphasis on symmetric groups \(S_n\).

As the starting point, let us try to classify all representations of \(S_2\) over complex numbers. Suppose that \(V\) is a complex representation of \(S_2\). The group \(S_2\) is generated by one element \(\sigma=(12)\) and therefore to find subrepresentations of \(V\) it is sufficient to find subspaces of \(V\) that are invariant under the action of \(\sigma\). First consider the action of \(\sigma\) on \(V\). Let \(v\) be an eigenvector of this action, with the eigenvalue \(\theta\). Then the subspace spanned by \(v\) is invariant under the action of \(S_2\), and therefore all representations of \(S_2\) must be one-dimensional. Moreover, since \(\sigma^2\) is the identity element of \(A_2\), then \(\theta^2=1\) which imples that \(\theta=\pm 1\). There are only two irreducible complex representations of \(S_2\):

• when \(\theta=1\) then we get the trivial representation

• when \(\theta=-1\) then we get the alternating representation

We can perform a similar calculation for \(S_3\). The group \(S_3\) is generated by \(\sigma=(123)\) and \(\tau=(12)\). Let \(V\) be a complex representation of \(S_3\), then we need to find all subspaces of \(V\) invariant under the action of both \(\sigma\) and \(\tau\). Let \(v\) be the eigenvector for the action of \(\sigma\), with the eigenvalue \(\theta\). Let \(w=\tau.v\). Because we know that \(\tau \sigma\tau=\sigma^2\) in \(S_3\) then

\[ \sigma.w=\sigma\tau.v=\tau\sigma^2v=\tau.(\theta^2)v=\theta^2\tau.v=\theta^2w \]

and therefore \(w\) is also an eigenvector of \(\sigma\) with the eigenvalue \(\theta^2\). It follows that the subspace generated by \(v\) and \(w\) is invariant under \(\sigma\) and \(\tau\), hence under all of \(S_3\). In particular, an irreducible complex representation of \(S_3\) over the complex numbers can have dimension no hogher than two.

Now suppose that \(V\) is irreducible. Since \(\sigma^3\) is the identity element of \(S_3\), the eigenvalue \(\theta\) must satisfy \(\theta^3=1\). There are two cases to consider:

• \(\theta=1\). Then \(w=v\) because otherwise the subspace spanned by \(v+w\) would be a one-dimensional invariant subspace of \(V\), and therefore \(V\) would not be irreducible. Then \(V\) is one-dimensional and we know that \(\sigma\) acts trivially on it. Now, since \(\tau^2=e\) we have only two options:

• \(\tau\) acts trivially: \(V\) is the trivial representation

• \(\tau\) acts as \(-1\): \(V\) is the alternating representation

• \(\theta\neq 1\). In this case \(\theta\neq \theta^2\), so the vectors \(v\) and \(w\) have distinct eigenvalues \(\theta\) and \(\theta^2\). Moreover, since \(\theta^3=1\) ther \(\theta^2+\theta+1=0\). Then, the map

\[\begin{split} \begin{align*} &V\to W=\{(x_1,x_2,x_3):\sum x_i=0\}\subset \mathbb{C}^3\\ &v\mapsto (1,\theta,\theta^2),\qquad w\mapsto (\theta,1,\theta^2) \end{align*} \end{split}\]

defines an isomorphism from \(V\) to the standard representation of \(S_3\).

To conclude, there are only 3 complex irreducible representations of \(S_3\):

• trivial (one-dimensional) representation

• alternating (one-dimensional) representation

• standard (two-dimensional) representation