Which of the following representations are irreducible?
1) The tautological representation of $D_n$ on $\mathbb{R}^2$
2) The action of $U(1)$ on $\mathbb{C}$ by multiplication
3) The tautological action of $GL(V)$ on $V$
4) The group homomorphism $(\mathbb{Q}, +) \longrightarrow GL(\mathbb{Q}^2)$ given by $\lambda \mapsto \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$
5) The permutation representation of $S_n$ on $\mathbb{C}^n$
6) The regular representation of $\mathbb{Z}_4$
7) The action of $SL_2(\mathbb{R})$ on the space of all $2 \times 2$ real matrices by left multiplication
8) The action of $SL_2(\mathbb{R})$ on the space of all $2 \times 2$ real matrices by conjugation
Here are my responses. Can someone please verify them? (Note: This is not homework!)
1) This is irreducible, since any line passing through the origin is not mapped to itself by a rotation by an angle other than $\pi$.
2) This is irreducible, because any line passing through the origin is not mapped to itself by multiplication by an element $z \in U(1)$ with phase not equal to $0$ or $\pi$.
3) My intuition tells me that this is irreducible
4) This is reducible. Note that the span of the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$ is a subrepresentation.
5) This is reducible, since $\left\lbrace \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix} : \sum x_i = 0 \right\rbrace$ is a subrepresentation (of dimension $n-1$).
6) This is reducible, since $\operatorname{span} \left\lbrace \begin{bmatrix} 1 \\ 1 \\ 1 \\ 1 \end{bmatrix} \right\rbrace$ is a subrepresentation
7) My intuition tells me that this is irreducible
8) This is reducible, since $\operatorname{span} \left\lbrace \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \right\rbrace$ is a subrepresentation.
Best Answer
Here are some remarks that should help you to improve your answers,first it seems that in each question you have some field you are working on, maybe it would be good to highlight this more clearly because the answers may change depending on the field...
1)(Field $\mathbb{R}$) does there always exist a rotation with an angle different from $\pi$? (think about the case $n=2$).
2) Here you seem to be working over $\mathbb{R}$ if this is the case, you should write $\mathbb{R}^2$ rather than $\mathbb{C}$. If you are working over $\mathbb{R}$ your argument is ok, if you are working over $\mathbb{C}$ it is a $1$-dimensional representation (hence irreducible).
3) The action of $GL(V)$ over $V-\{0\}$ is transitive (justify this and then deduce the irreducibility).
4) Ok.
5) Ok.
6) Ok.
7) I do not think so. Take the following subspaces :
$$V_1:=\{\begin{pmatrix}0&*\\0&*\end{pmatrix}\}\text{ and } V_2:=\{\begin{pmatrix}*&0\\*&0\end{pmatrix}\} $$
We have $M_{2,2}(\mathbb{R})=V_1\oplus V_2$, what can we say about the action of $M_{2,2}(\mathbb{R})$ regarding this decomposition?
8) Ok.