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\title{Homework 7}

\maketitle

\vglue-30pt

\subsection*{Dedekind Domains}

\bigbreak\noindent
{\textbf{Problem 1.}} Let $A$ be a Dedekind domain and let $\mathfrak{a},
\mathfrak{b}$ be nonzero ideals. We say that $\mathfrak{a}$ {\textit{divides}}
$\mathfrak{b}$ and write $\mathfrak{a}|\mathfrak{b}$ if
$\mathfrak{b}=\mathfrak{a}\mathfrak{c}$ for some ideal $\mathfrak{c}$. Show
that $\mathfrak{a}|\mathfrak{b}$ if and only if $\mathfrak{a} \supseteq
\mathfrak{b}$.

\bigbreak\noindent
{\textbf{Problem 2}}. Let $K / F$ be a finite separable extension, and let
$A$ be a Dedekind domain whose field of fractions is $F$. Let $B$ be the
integral closure of $A$ in $K$. By Problem~2 in Homework~4, $B$ is also a
Dedekind domain. Let $\mathfrak{P}$ be a maximal ideal of $B$. Show that
$\mathfrak{P} \cap A$ is a nonzero prime ideal of $A$. Furthermore if
$\mathfrak{p}$ is a maximal ideal of $A$, prove that $\mathfrak{P} \cap A
=\mathfrak{p}$ if and only if $\mathfrak{P}|\mathfrak{p}B$.

\subsection*{Group representations}

Let $G$ be a group. Recall that the {\textit{commutator subgroup}} or
{\textit{derived group}} $G'$ is the subgroup generated by commutators $[x, y] =
x y x^{- 1} y^{- 1}$. As we discussed in class, it is a normal subgroup and $G
/ G'$ is abelian. Moreover any homomorphism $G \longrightarrow A$, where $A$
is an abelian group, factors uniquely through the quotient $G / G'$.

Also recall that the characters of the one-dimensional representations of $G$
are called {\textit{linear characters}}. It is easy to see that these are just
the homomorphisms $G \longrightarrow \mathbb{C}^{\times}$. Thus every linear
character factors through $G / G'$.

\bigbreak\noindent
{\textbf{Problem 3}}. Let $G$ be a nonabelian group of order $21$ with
presentation
\[ G = \langle x, y|x^7 = y^3 = 1, y x y^{- 1} = x^2 \rangle . \]
Determine the conjugacy classes and give a representative $g_i$ for each. To
describe a character $\chi$ of $G$ it is sufficient to tell us $\chi (g_i)$
for each conjugacy class representatives. Let $Q = \langle x \rangle$ be the
7-Sylow subgroup, which is normal. Show that $Q = G'$ and determine the linear
characters of $G$. Use this information to determine the total number of
irreducible representations and their degrees.

\bigbreak\noindent
{\textbf{Problem 4.}} Continuing from Problem 3, note that if $\zeta$ is a
7-th root of unity and
\[ \xi = \left(\begin{array}{ccc}
     \zeta &  & \\
     & \zeta^2 & \\
     &  & \zeta^4
   \end{array}\right), \qquad \eta = \left(\begin{array}{ccc}
     & 1 & \\
     &  & 1\\
     1 &  & 
   \end{array}\right) \]
then $\xi^7 = \eta^3 = I$ (the identity matrix) and $\eta \xi \eta^{- 1} =
\xi^2$. Use this information to construct an irreducible representation of $G$
and finish computing the character table.

If $\mathcal{V}$ is the category of finite dimensional vector spaces and
$\mathcal{F}$ is a functor from $\mathcal{V}$ to itself, we may apply
$\mathcal{F}$ to a representation $\pi : G \longrightarrow \tmop{GL} (V)$:
defining
\[ (\mathcal{F} \pi) (g) =\mathcal{F} (\pi (g)) : \mathcal{F}V \longrightarrow
   \mathcal{F}V \]
gives a representation $\mathcal{F} \pi : G \longrightarrow \tmop{GL}
(\mathcal{F}V)$. In the next exercise we consider the functors of tensor
square, exterior square and symmetric square, which we will denote
$\otimes^2$, $\wedge^2$ and $\vee^2$.
\bigbreak\noindent
{\textbf{Problem 5.}} Let $V$ be a $\mathbb{C} [G]$-module, and let $\pi :
G \longrightarrow \tmop{GL} (V)$ be a representation. Let $\chi : G
\longrightarrow \mathbb{C}$ be the character of $\pi$. Show that the
characters of $\otimes^2 \pi$, $\wedge^2 \pi$ and $\vee^2 \pi$ are
\[ \otimes^2 \chi (g) = \chi (g)^2, \qquad \wedge^2 \chi (g) = \frac{1}{2}
   (\chi (g)^2 - \chi (g^2)), \qquad \vee^2 \chi (g) = \frac{1}{2} (\chi (g)^2
   + \chi (g^2)) . \]
{\textbf{Hint:}} Express these in terms of the eigenvalues of $\pi (g)$.

\bigbreak\noindent
{\textbf{Problem 6.}} Let $\pi : G \longrightarrow \tmop{GL} (V)$ be an
irreducible finite-dimensional complex representation of the finite group $G$. Note that
$\pi$ extends to a $\mathbb{C}$-algebra homomorphism $\mathbb{C} [G]
\longrightarrow \tmop{End} (V)$, also denoted $\pi$. Let $\mathcal{Z}$ be the
center of $\mathbb{C} [G]$. Show that there exists a $\mathbb{C}$-algebra
homomorphism $\omega_{\pi} : \mathcal{Z} \longrightarrow \mathbb{C}$ such that
$\pi (\xi) v = \omega_{\pi} (\xi) v$ for $\xi \in \mathcal{Z}$, called the
{\textit{central character}} of $\pi$. Let $g \in G$ and let $\mathcal{C}$ be the
conjugacy class of $g$. Let $\mathcal{C}$ be a conjugacy class of $G$. Observe
that $\mathfrak{C}= \sum_{h \in \mathcal{C}} h \in \mathcal{Z}$. Prove that
\[ \omega_{\pi} (\mathfrak{C}) = \frac{\chi (g) | \mathcal{C} |}{\chi (1)} .
\]
where $\chi$ is the character of $\pi$.

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