2 Character theory 2.2 Orthogonality of characters 2.4 Inner products and homomorphisms

2.3 Example: S₄

Example 2.20.

We determine the character table of $S_{4}$ . First, we have the trivial representation $\mathbbm{1}$ and the sign character $\epsilon$ :

\begin{array}[h]{c|rrrrr}&e&(12)&(12)(34)&(123)&(1234)\\ &1&6&3&8&6\\ \hline\cr\mathbbm{1}&1&1&1&1&1\\ \epsilon&1&-1&1&1&-1.\end{array}

Next, we consider the permutation representation $V$ of $S_{4}$ on $\{1,2,3,4\}$ . This contains a copy of the trivial representation, so we write $V=\mathbbm{1}\oplus W$ for some representation $W$ , whose character $\chi$ satisfies

\chi(g)=\#\{\text{fixed points of $g$}\}-1.

Notice that we also have an operation on representations known as twisting: if $(\rho,V)$ is an irreducible representation of $G$ with character $\chi$ and $\psi$ is a one-dimensional character of $G$ , then we can define a new representation, $\rho\psi$ , of $G$ on $V$ by the formula $(\rho\psi)(g)=\rho(g)\psi(g)$ . It has character $\chi\psi$ . In this case, we can look at $\chi\epsilon$ and see that it will also be the character of an irreducible representation.

\begin{array}[h]{c|rrrrr}&e&(12)&(12)(34)&(123)&(1234)\\ &1&6&3&8&6\\ \hline\cr\chi&3&1&-1&0&-1\\ \chi\epsilon&3&-1&-1&0&1\\ \end{array}

Since $\left\langle\chi,\chi\right\rangle=\frac{1}{24}(3^{2}+6\times 1^{2}+3\times(-1% )^{2}+8\times 0^{2}+6\times(-1)^{2})=1$ , $\chi$ is irreducible. Note also that $\chi\otimes\epsilon$ is different to $\chi$ , and also irreducible.

There is one more row of the table to find. This can be done using orthonormality with the first column, remembering that the dimension must be a positive integer (in this case, it must be two): we obtain that the full character table is

\begin{array}[h]{c|rrrrr}&e&(12)&(12)(34)&(123)&(1234)\\ &1&6&3&8&6\\ \hline\cr\mathbbm{1}&1&1&1&1&1\\ \epsilon&1&-1&1&1&-1\\ \chi&3&1&-1&0&-1\\ \chi\epsilon&3&-1&-1&0&1\\ \psi&2&0&2&-1&0\end{array}

Notice that we constructed the character of the final representation without constructing the representation itself!

2.3.1 Characters of lifts

Recall that if $K\subset G$ is a normal subgroup then we can lift representations of $G/K$ to representations of $G$ , as in Section 1.2.3.

Lemma 2.21.

If $\chi$ is the character of a representation of $G/K$ and $\tilde{\chi}$ is the character of its lift to $G$ , then

\tilde{\chi}(g)=\chi(gK)

for all $g\in G$ .

Proof.

Immediate from the definition. ∎

Example 2.22.

In exercise 1.29, we explained how to lift the irreducible two-dimensional representation of $S_{3}$ to a representation of $S_{4}$ . Then its character agrees with that called $\psi$ in the character table of $S_{4}$ .

We can recognise the kernel of a representation using its character.

Proposition 2.23.

Let $\rho$ be a representation of $G$ with character $\chi$ and dimension $d$ . Then

\ker(\rho)=\{g\in G:\chi(g)=d\}.

Proof.

It is clear that $g\in\ker(\rho)$ then $\chi(g)=\operatorname{tr}I=d$ . We leave the converse as an exercise. ∎

2.3.2 Decomposing a representation

Example 2.24.

We apply the method of Theorem 2.19 to decompose a naturally occuring representation of $S_{4}$ . Let

X=\{S\subset\{1,2,3,4\}:|S|=2\}.

Then $S_{4}$ acts on $X$ . If we think of $S_{4}$ as the symmetry group of the tetrahedron with vertices labeled 1, 2, 3, 4, then this is ‘the same as’ the action on the set of edges (each edge being determined by a set of two vertices).

Let $(\rho,V)$ be the permutation representation obtained from $X$ and let $\chi_{\rho}$ be its character. Then

\chi_{\rho}(g)=|X^{g}|=|\left\{\{x,y\}\subset\{1,2,3,4\}:\{gx,gy\}=\{x,y\},x% \neq y\right\}|.

We have:

\begin{array}[h]{c|rrrrr}&e&(12)&(12)(34)&(123)&(1234)\\ &1&6&3&8&6\\ \hline\cr\chi_{\rho}&6&2&2&0&0\end{array}

Then we compute $\left\langle\chi_{\rho},\mathbf{1}\right\rangle=\left\langle\chi_{\rho},\chi% \right\rangle=\left\langle\chi_{\rho},\psi\right\rangle=1$ , $\left\langle\chi_{\rho},\epsilon\right\rangle=\left\langle\chi_{\rho},\chi% \epsilon\right\rangle=0$ . It follows that

\chi_{\rho}=\mathbbm{1}+\chi+\psi.

\rho\cong W_{0}\oplus W_{1}\oplus W_{2}

where $W_{0}=\mathbb{C}$ is the trivial representation, $W_{1}$ is isomorphic to the 3-D representation of $S_{4}$ with character $\chi$ , and $W_{2}$ is isomorphic to the unique 2-D representation of $S_{4}$ (which has character $\psi$ ).

Exercise 2.25.

In the previous example, find $W_{0}$ , $W_{1}$ , $W_{2}$ as subspaces of $V$ .

Remark 2.26.

The problem of finding irreducible representations of $G$ such that $V$ is isomorphic to their direct sum is called “decomposing $V$ into irreducibles representations (or irreps, or irreducibles)”. It can be solved with character theory as in example 2.24.

The problem of finding irreducible subrepresentations of $V$ such that it is their (internal) direct sum is called “decomposing $V$ into irreducible subrepresentations” and can require more understanding of the nature of the representation $V$ . Exercise 2.25 is an example of this.

2.3 Example: S4

Example 2.20.

2.3.1 Characters of lifts

Lemma 2.21.

Proof.

Example 2.22.

Proposition 2.23.

Proof.

2.3.2 Decomposing a representation

Example 2.24.

Exercise 2.25.

Remark 2.26.

2.3 Example: S₄