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\title{Math 122: Homework 5}
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\begin{itemize}[noitemsep]
  \item Section 18.3 \# 6,7
	\item Section 19.1 \#2,3,7,8,9
\end{itemize}

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{\textbf{Section 18.3 \#6.}} Let $\varphi : G \longrightarrow \operatorname{GL} (V)$
be a representation with character $\psi$. Let $W$ be the subspace
\[ W = \left\{ v \in V| \text{$\varphi (g) v = v$ for all $g \in G$} \right\}
\]
fixed pointwise by all elements of $G$. Prove that $\dim (W) = (\psi, \chi_1)$
where $\chi_1$ is the principal character of $G$.

\

{\textbf{Section 18.3 \#7.}} Let Assume that $V$ is a $\mathbb{C} [G]$-module
on which $G$ acts by permuting the basis $\mathcal{B}= \{ e_1, \cdots, e_n
\}$. Write $\mathcal{B}$ as a disjoint union of orbits $\mathcal{B}_1, \cdots,
\mathcal{B}_n$ of $G$ on $\mathcal{B}$.

\medbreak
(a) Prove that $V$ decomposes as a $\mathbb{C} [G]$-module as $V_1 \oplus
\cdots \oplus V_t$ where $V_i$ is the span of $\mathcal{B}_i$.

\medbreak
(b) Prove that if $v_i$ is the sum of the vectors in $\mathcal{B}_i$, then
$v_i$ is the unique $\mathbb{C} [G]$-submodule of $V_i$ affording the trivial
representation (in other words, any vector in $V_i$ that is fixed under the
action of $G$ is a multiple of $v_i$. [Use the fact that $G$ is transitive on
$\mathcal{B}_i$. See also Exercise 8 in Section 1.1.

\medbreak
(c) Let $W = \left\{ v \in V| \text{$\varphi (g) v = v$ for all $g \in G$}
\right\}$. Prove that $\dim (W) = t$ is the number of orbits of $G$
on~$\mathcal{B}$.

\

{\textbf{Section 19.1 \#2.}} Compute the degrees of the irreducible
characters of $D_{16}$.

\

{\textbf{Section 19.1 \#3.}} Compute the degrees of the irreducible
characters of $A_5$. Deduce that the degree $6$ irreducible representation is
not irreducible when restricted to $A_5$. [The conjugacy classes of $A_5$ are
worked out in Section 4.3.

\

{\textbf{Section 19.1 \#7.}} Show that $S_6$ has an irreducible character of
degree~$5$.

\

{\textbf{Section 19.1 \#8.}} Calculcate the character table of $D_{10}$.
(This table contains nonreal entries.)

\

{\textbf{Section 19.1 \#9.}} Calculcate the character table of $D_{12}$.

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