Let $G$ be a finite group. Suppose that $\rho: G \rightarrow G \mathrm{G}(V)$ is a finite-dimensional complex representation of dimension $d$. Let $n \in \mathbb{N}$ be arbitrary.

(i) Define the $n$th symmetric power $S^{n} V$ and the $n$th exterior power $\Lambda^{n} V$ and write down their respective dimensions.

Let $g \in G$ and let $\lambda_{1}, \ldots, \lambda_{d}$ be the eigenvalues of $g$ on $V$. What are the eigenvalues of $g$ on $S^{n} V$ and on $\Lambda^{n} V$ ?

(ii) Let $X$ be an indeterminate. For any $g \in G$, define the characteristic polynomial $Q=Q(g, X)$ of $g$ on $V$ by $Q(g, X):=\operatorname{det}(g-X I)$. What is the relationship between the coefficients of $Q$ and the character $\chi_{\Lambda^{n}} V$ of the exterior power?

Find a relation between the character $\chi_{S^{n}} V$ of the symmetric power and the polynomial $Q$.