# Week 2 Exercises

Some questions we left open last time:

1. Let $\rho: \mathbb{Z}/n \to \mathbb{C}^*$ be given by $1 \mapsto \zeta$ for some $n$-th root of unity $\zeta$. Show that $\rho^*$ is $1 \mapsto \bar{\zeta}$. Conclude that $\rho, \rho^*$ are not isomorphic.
2. Let $A$ be an $n \times n$ matrix of finite order ($A^m = I$ for some $m \in \mathbb{Z}$). Show that $tr(A^{-1}) = \overline{tr(A)}$, without using eigenvalues.
3. In the proof of Prop. 1 of Serre (see Section 2.1), he states that “… $\rho_s$ can be defined to be a unitary matrix”. Show that every representation of a finite group is unitary. (As part of this exercise, give an appropriate definition of a unitary representation.)
4. Prove that the character of $V_1 \otimes V_2$ is given by $\chi_1 \cdot \chi_2$. If $\rho$ is a representation, prove that the character of a dual representation is $\chi_{\rho^*} = \overline{\chi_\rho}$. Find the characters of $\textrm{Sym}^2(\rho)$ and $\rho \wedge \rho = \textrm{Alt}^2(\rho)$.
5. If $f: (\rho_1, V_1) \to (\rho_2, V_2)$ is a map of representations, show that the kernel and image of $f$ are subrepresentations of $(\rho_1, V_1), (\rho_2, V_2)$ respectively.
6. Midterm question: Give your own proof of Serre Thm. 3 (section 2.3), that characters of irreducible representations form an orthonormal system.
7. Prove Prop. 6 of Serre (section 2.5): If $(\rho, V)$ is an irreducible representation on $G$ of degree $n$ and character $\chi$, and $f$ is a class function on $G$, then $\rho_f := \sum_{t \in G} f(t)\rho(t)$ is a homothety of ratio $\frac{1}{n} \sum_{t \in G} f(t) \chi(t)$.
8. Prove that the number of degree 1 representations on a finite group $G$ is $|G^{ab}|$.