8 Problems for Michaelmas 8.1 Problems for section 1 8.3 Problems for section 3

8.2 Problems for section 2

Problem 19.

Find the character tables of the following groups (you shouldn’t need to use orthogonality for these). Note that you will have to find the conjugacy classes!

1.

$C_{4}$
2.

$C_{3}\times C_{3}$
3.

$D_{4}$
4.

$D_{5}$
5.

$Q_{8}$
6.

$D_{n}$ .

Solution

Problem 20.

Let $G$ be a finite group acting on a finite set $X$ . Let $\chi$ be the character of the permutation representation. Prove that

\chi(g)=|\{x\in X:gx=x\}|.

Find the character of the regular representation.

Solution

Problem 21.

The center $Z(\mathbb{C}[G])$ of the group ring $\mathbb{C}[G]$ is the set of elements

\sum_{g\in G}a_{g}[g]\in\mathbb{C}[G]

which commute with all other elements of the group ring (it is enough to check that they commute with all elements $[g]$ for $g\in G$ ).

Show that $\sum_{g\in G}a_{g}[g]$ is in $Z(\mathbb{C}[G])$ if and only if the function $g\mapsto a_{g}$ is a class function.

Solution

Problem 22.

Let $(\rho,V)$ be a representation of $G$ with character $\chi$ and dimension $d$ . Show that

|\chi(g)|\leq d

for all $g\in G$ with equality if and only if $\rho(g)$ is a scalar matrix. Deduce that

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

Solution

Problem 23.

1.

Find the character table of $A_{4}$ .
2.

For each irreducible representation of $S_{4}$ , decompose its restriction to $A_{4}$ into irreducibles. ⁷⁷ 7 When we say decompose $V$ into irreducible subrepresentations we mean find actual subrepresentations of $V$ such that $V$ is their direct sum. When I say decompose $V$ into irreducible representations, or just irreducibles, I just mean find irreducible representations such that $V$ is isomorphic to their direct sum — you don’t have to say how they live inside $V$ .

Solution

Problem 24.

Do exercise 2.25: if $V$ is the permutation representation attached to $S_{4}$ acting on the set of edges of the tetrahedron, find the three irreducible subrepresentations of $V$ . ⁷ Solution

Problem 25.

Let $W\subset V$ be finite-dimensional vector spaces and let $\pi:V\rightarrow W$ be a projection. Show that

\operatorname{tr}(\pi)=\dim W.

Solution

Problem 26.

Let $G$ act on a set $X$ , and let $(\pi,V)$ be the associate permutation representation with character $\chi_{\pi}$ .

1.

Show that $\dim V^{G}$ is the number of orbits of $G$ acting on $X$ .
2.

By considering $\left\langle\mathbbm{1},\chi_{\pi}\right\rangle$ , prove Burnside’s lemma: the number of orbits on $X$ is the average number of fixed points of elements of $G$ .

Solution

Problem 27.

Consider the representation $\rho$ of $S_{3}$ on $V=\mathbb{C}^{2}$ given by

\rho(12)=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\qquad\rho(123)=\begin{pmatrix}\omega&0\\ 0&\omega^{2}\end{pmatrix}

with $\omega=e^{2\pi i/3}$ .

1.

Write down the matrices of $(\rho\otimes\rho)(12)$ and $(\rho\otimes\rho)(123)$ with respect to the basis

$e_{1}\otimes e_{1},e_{1}\otimes e_{2},e_{2}\otimes e_{1},e_{2}\otimes e_{2}$

of $V\otimes V$ (where $e_{1}$ and $e_{2}$ are the standard basis of $V$ ).
2.

Write the character of $V\otimes V$ as a sum of irreducible characters.
3.

For each of the irreducible characters of $V\otimes V$ used in the previous part, find a subrepresentation of $V\otimes V$ with that character.
4.

Find a $G$ -isomorphism $\epsilon\otimes V\rightarrow V$ . Find a $G$ -isomorphism $V^{*}\rightarrow V$ .
5.

Let $V^{\otimes n}=V\otimes\ldots\otimes V$ with $n$ factors. Decompose $V$ into irreducible representations. ⁷

Solution

Problem 28.

Exercise 2.40. If $V=\mathbb{C}^{2}$ , prove that $e_{1}\otimes e_{2}+e_{2}\otimes e_{1}\in V\otimes V$ cannot be written in the form $v\otimes w$ . Solution

Problem 29.

Let $V$ be an irreducible five-dimensional representation of $S_{5}$ . Decompose $\operatorname{Sym}^{2}V$ and $\Lambda^{2}V$ into irreducible representations.

Solution

Problem 30.

Let $V=\mathbb{C}^{4}$ be the permutation representation of $S_{4}$ , and let $W$ be the two-dimensional irreducible representation of $S_{4}$ .

1.

Show that $\operatorname{Sym}^{2}V$ has a unique subrepresentation isomorphic to $W$ .
2.

Use the projection operator to find that subrepresentation.

Solution

Problem 31.

Using the character table of $S_{5}$ , find the character table of $A_{5}$ . Using the character table, show that $A_{5}$ is simple (that is, it has no nontrivial proper normal subgroups). Hint: every element of $A_{5}$ is conjugate to its inverse. Why does this imply the character values are all real? Solution

Problem 32.

A group $G$ of order $168$ has conjugacy classes $C_{1}$ , $C_{2}$ , $C_{3}$ , $C_{4}$ , $C_{7A}$ and $C_{7B}$ where each conjugacy class is labelled by the order of any element in that class (so, for example, any element of $C_{7A}$ or $C_{7B}$ has order $7$ ). The following shows one of the rows of the character table of $G$ .

\begin{array}[]{c|cccccc}\text{class}&C_{1}&C_{2}&C_{3}&C_{4}&C_{7A}&C_{7B}\\ \text{size}&1&21&56&42&24&24\\ \hline\cr\chi&3&-1&0&1&\frac{-1+\sqrt{-7}}{2}&\frac{-1-\sqrt{-7}}{2}\\ \end{array}

(8.1)

1.

Show that, if $x$ is an element of $C_{7A}$ or $C_{7B}$ , then $x$ is conjugate to $x^{2}$ .
2.

Find the character table of $G$ . Assume the result of the first part if you were not able to prove it.

Solution

Problem 33.

(challenge) Show that, if $V$ is any faithful representation of $G$ and $W$ is an irreducible representation of $G$ , then $W$ is isomorphic to a subrepresentation of $V^{\otimes n}$ for some $n\geq 1$ .