Character of representation of $D_5$ on $\mathbb{R}^5$.

Question

How does one stablish the character of a representation of $D_5$ on $\mathbb{R}^5$? The problem lies in the fact that I am unsure as to what the rotation and reflection matrices will look like in dimensions $> 3$, so cannot calculate the traces. Any help is very much appreciated.

Answer 1

Just write down the matrices.

• For the identity, $\iota (a_1, a_2, a_3, a_4, a_5) = (a_1, a_2, a_3, a_4, a_5)$, the matrix is $$ \begin{bmatrix} 1&0&0&0&0 \\ 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&0&0&1&0 \\ 0&0&0&0&1 \end{bmatrix}$$ whose trace is $5$.

• For the rotation $r(a_1, a_2, a_3, a_4, a_5) = (a_2, a_3, a_4, a_5, a_1)$, the matrix is $$ \begin{bmatrix} 0&0&0&0&1 \\ 1&0&0&0&0 \\ 0&1&0&0&0 \\ 0&0&1&0&0 \\ 0&0&0&1&0 \end{bmatrix}$$ whose trace is $0$. (The other four rotations have the same trace.)

• For the reflection $s(a_1, a_2, a_3, a_4, a_5) = (a_5, a_4, a_3, a_2, a_1)$, the matrix is $$ \begin{bmatrix} 0&0&0&0&1 \\ 0&0&0&1&0 \\ 0&0&1&0&0 \\ 0&1&0&0&0 \\ 1&0&0&0&0\end{bmatrix}$$ whose trace is $1$. (The other five rotations have the same trace.)

Finally, I'll remark that, although the elements of the group $D_5$ can be thought of as rotations and reflections of a pentagon, they are not acting as rotations and reflections in this representation on $\mathbb R^5$. An actual reflection in $\mathbb R^5$ would have eigenvalues $1, -1, -1, -1, -1$, and hence a trace of $-3$.