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\begin{document}

\title{Homework 8}

\maketitle

\subsection*{Dedekind Domains}

We recall that a field $k$ is {\textit{perfect}} if every finite extension is
separable. For example, finite fields, algebraically closed fields and fields
of characteristic zero are all perfect. The field $\mathbb{F}_p (X)$ of
fractions of the polynomial ring $\mathbb{F}_p [X]$ is not perfect, since the
extension $\mathbb{F}_p (X^{1 / p}) /\mathbb{F}_p (X)$ is inseparable.

\bigbreak\noindent
{\textbf{Problem 1.}} Let $E / F$ be a finite separable field extension.
Let $A$ be a Dedekind domain with field of fractions $F$ and let $B$ be the
integral closure of $A$ in $E$. Thus $B$ is a Dedekind domain. Assume that $B
= A [\alpha]$ for some $\alpha \in B$. Let $\mathfrak{p}$ be a maximal ideal
of $A$. Assume that $\overline{A} = A /\mathfrak{p}$ is a perfect field. We
know that $\mathfrak{p}B$ can be factored into maximal ideals:
\[ \mathfrak{p}B =\mathfrak{P}_1^{e_1} \cdots \mathfrak{P}_r^{e_r} . \]
The quotient ${\overline{B}}_i = B /\mathfrak{P}_i$ is an extension of
$\overline{A}$. Let $n = [E : F]$ and let $f_i = \left[ {\overline{B}}_i:
\overline{A} \right]$. Prove that
\[ \sum_{i = 1}^r f_i \leqslant n. \]
{\textbf{Hint:}} Let $\Omega$ be an algebraic closure of $A /\mathfrak{p}$.
Count the homomorphisms $B \longrightarrow \Omega$ extending the canonical map
$A \longrightarrow A /\mathfrak{p}$. You may need the assumption that
$\overline{A} = A /\mathfrak{p}$ to do the counting correctly.

\medbreak\noindent
{\textbf{Remark:}} The statement is true without the simplifying assumption
$B = A [\alpha]$. In fact, the correct statement is
\[ \sum_{i = 1}^r e_i f_i = n. \]
See Lang, {\textit{Algebraic Number Theory}}, Proposition~21 in Chapter I (page
24). This result is basic in algebraic number theory, and also in the study of
algebraic curves. The numbers $f_i$ and $e_i$ are called the {\textit{residue
class degree}} and {\textit{ramification index}}. Ramification is rare in the
sense that all $e_i = 1$ for all but finitely many primes $\mathfrak{p}$ of
$A$. Indeed, assuming $B = A [\alpha]$ it is easy to see that all $e_i = 1$
unless $\mathfrak{p}$ divides the discriminant of the irreducible polynomial
satisfied by $\alpha$.

\subsection*{Group Representations}

\bigbreak\noindent
{\textbf{Problem 2.}} Let
\[ G = \langle x, y|x^7 = y^3 = 1, y x y^{- 1} = x^2 \rangle \]
be the nonabelian group of order $21$. The cyclic group $N = \langle x
\rangle$ has $7$ linear characters. Compute the induced character
$\tmop{Ind}_N^G (\chi)$ for each of these.

\bigbreak\noindent
{\textbf{Problem 3.}} Let $(\pi, V)$ be an irreducible representation of
the finite group $G$, with character $\chi$. Let $\mathcal{C}$ be a conjugacy
class of $G$ and let $g \in \mathcal{C}$. Prove that
\[ | \mathcal{C} | \frac{\chi (g)}{\chi (1)} \]
is an algebraic integer, i.e. and element of $\overline{\mathbb{Q}}$ that is
integral over $\mathbb{Z}$.

\medbreak\noindent
{\textbf{Hint:}} Let $\mathcal{C}_1, \cdots, \mathcal{C}_h$ be the
conjugacy classes of $G$, and let $g_i$ be a representative of each
$\mathcal{C}_i$. Let $\mathfrak{C}_i = \sum_{g \in \mathcal{C}_i} g$; it is
easy to see that these elements span the center of $\mathbb{C} [G]$. Deduce
that the quantities $| \mathcal{C}_i | \chi (g_i) / \chi (1)$ span a finitely
generated $\mathbb{Z}$-subalgebra of $\mathbb{C}$.

\bigbreak\noindent
{\textbf{Problem 4.}} Let $G$ be a finite group and $H$ a subgroup of index
two. Let $(\pi, V)$ be an irreducible representation of $H$. Prove that either
$\tmop{Ind}_H^G (\pi)$ is irreducible, or $\pi$ can be extended to an
irreducible representation of $G$. And in the second case, show that there are
two such extensions.

\bigbreak\noindent
{\textbf{Problem 5.}} Let $G$ be a finite group and let $V$ be an
irreducible $\mathbb{C} [G]$-module, with character $\chi$. Define the
{\textit{Frobenius-Schur indicator}}
\[ \varepsilon (\chi) = \frac{1}{| G |} \sum_{g \in G} \chi (g^2) . \]
Prove that $\varepsilon (\chi) = 0$ unless $\chi$ is real-valued, in which
case it equals $\pm 1$. Then $\varepsilon (\chi) = 1$ if the symmetric square
module has a nonzero invariant vector (i.e.~$\dim (\vee^2 V)^G = 1$) and
$\varepsilon (\chi) = - 1$ if $\dim (\wedge^2 V)^G = 1$.

Mackey theory is concerned with intertwining operators (i.e.~$\mathbb{C}
[G]$-module homomorphisms) between induced representations. It is a powerful
tool, and often exactly what is needed for some problem. Here is one version.

\begin{theorem}[Mackey]
  Let $G$ be a finite group and let $H_1$, $H_2$ be subgroups. Let $(\pi_1,
  V_1)$ and $(\pi_2, V_2)$ be representations of $H_1$, $H_2$ respectively. If
  $\gamma \in G$ and let $H_{\gamma} = H_2 \cap \gamma H_1 \gamma^{- 1}$.
  Define a representation $\pi_{\gamma}$ of $H_{\gamma}$ by
  \[ \pi_{\gamma} (h) = \pi_1 (\gamma^{- 1} h \gamma) . \]
  Then as vector spaces
  \[ \tmop{Hom}_{\mathbb{C} [G]} (\tmop{Ind}_{H_1}^G (\pi_1),
     \tmop{Ind}_{H_2}^G (\pi_2)) \cong \bigoplus_{\gamma \in H_2 \backslash G
     / H_1} \tmop{Hom}_{\mathbb{C} [H_{\gamma}]} (\pi_{\gamma}, \pi_2) . \]
  The notation means that we take a sum over a set of representatives $\gamma$
  for the set of double cosets $H_2 \gamma H_1$.
\end{theorem}

\begin{proof}
  We will prove this in the lectures, or see Lang's {\textit{Algebra}},
  Theorem~7.7 on page 695.
\end{proof}

\bigbreak\noindent
{\textbf{Problem 6.}} Let $F$ be a finite field. Let $G = \tmop{GL} (2, F)$
and let $B$ be the subgroup
\[ B = \left\{ \left(\begin{array}{cc}
     y_1 & x\\
     0 & y_2
   \end{array}\right) |y_1, y_2 \in F^{\times}, x \in F \right\} . \]
Let $\chi_1, \chi_2$ be characters of $F^{\times}$. Define a linear character
$\chi$ of $B$ by
\begin{equation}
  \label{chidef} \chi \left(\begin{array}{cc}
    y_1 & x\\
    0 & y_2
  \end{array}\right) = \chi_1 (y_1) \chi_2 (y_2)
\end{equation}
and let $\pi (\chi_1, \chi_2) = \tmop{Ind}_B^G (\chi)$. Now let $\psi_1$ and
$\psi_2$ be two more characters of $F^\times$. Prove that
\[ \dim \; \tmop{Hom} (\pi (\chi_1, \chi_2), \pi (\psi_1, \psi_2)) = A + B \]
where
\[ A = \left\{\begin{array}{ll}
     1 & \text{if $\chi_1 = \psi_1, \chi_2 = \psi_2$},\\
     0 & \text{otherwise},
   \end{array}\right. \qquad B = \left\{\begin{array}{ll}
     1 & \text{if $\chi_1 = \psi_2, \chi_2 = \psi_1$},\\
     0 & \text{otherwise} .
   \end{array}\right. \]
Deduce that $\pi (\chi_1, \chi_2)$ is irreducible if $\chi_1 \neq \chi_2$ and
isomorphic to $\pi (\chi_2, \chi_1)$.

\bigbreak

This problem shows how powerful Mackey's theorem is with a typical
application. Problem~5 constructs about half the irreducible representations
of $\tmop{GL} (2, F)$.

\end{document}
