\documentclass[11pt]{amsart}
\usepackage{amssymb}
\usepackage[all]{xy} 
% \include{mssymb}
\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}
\newtheorem{lemma}{Lemma}
\newtheorem{defn}{Definition}

\def\A{{\mathcal A}}
\def\B{{\mathcal B}}
\def\N{{\mathbb N}}
\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\cC{{\mathcal C}}
\def\F{{\mathbb F}}
\let\sec\S
\let\S\relax
\def\S{{\mathfrak S}}
\def\g{{\mathfrak g}}
\def\p{{\mathfrak p}}
\def\h{{\mathfrak h}}
\def\n{{\mathfrak n}}
\def\v{{\mathfrak v}}
\def\m{{\mathfrak m}}
\def\k{{\mathfrak k}}
\def\endproof{\hfill $\Box$}


\newcommand{\skipline}{\vspace{\baselineskip}}
\newcommand{\dis}{\displaystyle}
\newcommand{\noin}{\noindent}


% Mathematical notation


\newcommand{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\ra}{\rightarrow}
\newcommand{\pa}{\partial}
\newcommand{\gequ}{\geqslant}
\newcommand{\lequ}{\leqslant}
\newcommand{\Ext}{\mathrm{Ext}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Span}{\mathrm{Span}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\PGL}{\mathrm{PGL}}
\newcommand{\Aut}{\mathrm{Aut}}
\newcommand{\Coker}{\mathrm{Coker}}
\newcommand{\Ker}{\mathrm{Ker}}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\Id}{\mathrm{Id}}
\newcommand{\ov}{\overline}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\Tor}{\mathrm{Tor}}
\newcommand{\rk}{\mathrm{rk}}
\newcommand\bull{{\scriptscriptstyle \bullet}}

\begin{document}

\begin{center}
{\bf Math 601 -- Spring 2026 -- Harry Tamvakis}
\end{center}


\begin{center}
{\bf PROBLEM SET 10 -- Due April 30, 2026}
\end{center}



\skipline
\noin {\bf A1)} Suppose that $G$ is the infinite group
$\dis \left\{\left(\begin{array}{cc} 1 & n \\ 0 & 1\end{array}\right)\ \
|\ \ n\in\Z\right\}$
(note that $G$ is isomorphic to the additive group $(\Z,+)$.) Let
$V$ be the $\C G$-module $\C^2$, with the natural
multiplication by elements of $G$. Show that $V$ is not a direct
sum of irreducible representations of $G$ (this shows that Maschke's
theorem fails for infinite groups.)


  
\skipline
\noin
{\bf A2)} Suppose that
\[
G=D_4=\left<x,y\ |\ x^4=y^2=1, \ yx=x^{-1}y\right>.
\]
Check that there is a matrix representation $\rho:G\ra \GL_2(\C)$
such that
\[
\rho(x)=\left(\begin{array}{cc} -7 & 10 \\ -5 & 7\end{array}\right),
\ \
\rho(y)=\left(\begin{array}{cc} -5 & 6  \\ -4 & 5\end{array}\right).
\]
Find all $2\times 2$ matrices $A$ such that $A\rho(g)=\rho(g)A$ for
all $g\in G$. Hence determine whether or not $\rho$ is irreducible.

Do the same for the representation $\sigma$ of $G$, where
\[
\sigma(x)=\left(\begin{array}{cc} 5 & -6 \\ 4 & -5\end{array}\right),
\ \
\sigma(y)=\left(\begin{array}{cc} -5 & 6 \\ -4 & 5\end{array}\right).
\]



\skipline \noin {\bf A3)} For each $n\geq 2$, let $D_n$ be the
dihedral group of order $2n$. Show that $D_n$ has exactly two
one-dimensional representations, if $n$ is odd, and exactly four
one-dimensional representations, if $n$ is even.
   
  
\skipline
\noin
{\bf A4)} Consider the standard two dimensional representation of 
the dihedral group $D_n$ as symmetries of the regular $n$-gon. For which
values of $n\geq 3$ is it irreducible as a complex representation?

  
\skipline \noin {\bf A5)} Let $G$ be a finite group.  If $\rho: G\to
\GL_n(\C)$ is an irreducible matrix representation and $A$ is an
$n\times n$ matrix commuting with $\rho(g)$ for all $g\in G$, prove
that $A$ is a scalar matrix. [Hint: Show that the eigenspaces of $A$
  are $G$-stable.] Deduce that if $\rho$ is a faithful, irreducible,
complex representation then the center $Z$ of $G$ is cyclic and
$\rho(g)$ is a scalar matrix for all $g\in Z$.


\skipline \noin {\bf A6)} Recall from homework set \#7 that a complex
number $\lambda$ is an {\em algebraic integer} if it is the root of a
monic polynomial with integer coefficients.

\medskip
\noin
(a) Prove that $\lambda$ is an algebraic integer if and only if
$\lambda$ is an eigenvalue of some matrix $A$, all of whose
entries are integers.

\medskip
\noin
(b) If $\lambda$, $\mu$ are eigenvalues of matrices $A\in M_r(\Z)$ 
and $B\in M_s(\Z)$ respectively, prove that $\lambda\mu$ is an
eigenvalue of the matrix $A\otimes B$ and 
$\lambda +\mu$ is an eigenvalue of the matrix
$A\otimes I_s + I_r\otimes B$. Deduce that the
set of algebraic integers is a commutative ring.

\medskip
\noin
(c) Let $G$ be a finite group and $\chi$ a character of $G$. If 
$g\in G$, show that $\chi(g)$ is an algebraic integer. Deduce that
if $\chi(g)$ is a rational number, then $\chi(g)$ is an integer.

\skipline
\noin
{\bf A7)} Let $G$ be a finite subgroup of $\GL_n(\C)$. Prove that if $\dis\sum_{g\in G} \Tr(g)=0$, then $\dis\sum_{g\in G} g = 0$.
    
   




\skipline \noin {\bf B1)} Suppose $x_1,\ldots,x_n$ are $n$ variables. For each
$k$ with $1\leq k\leq n$ let 
$e_k(x_1,\ldots,x_n)$ be the $k$-th elementary symmetric
polynomial and define the $k$-th {\em power sum} by
\[
p_k(x_1,\ldots,x_n):=x_1^k+\cdots+x_n^k.
\]

\smallskip
\noin
(a) Prove {\em Newton's identities:}
\[
p_k-e_1p_{k-1}+e_2p_{k-2}-\cdots +(-1)^{k-1}p_1e_{k-1}+(-1)^kke_k=0
\]
for $1\leq k\leq n$.

\medskip
\noin
(b) For $1\leq k\leq n$, prove the determinantal formulas
\[
p_k=
\left|
\begin{array}{ccccc}
e_1 & 2e_2 & 3e_3 & \cdots & ke_k \\
1 & e_1 & e_2 & \cdots & e_{k-1} \\
0 & 1 & e_1 & \cdots & e_{k-2} \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & \cdots & \ \ \ 0 \ \ \ 1 & e_1
\end{array}
\right|
\]
and
\[
k!\,e_k =
\left|
\begin{array}{ccccc}
p_1 & p_2 & p_3 & \cdots & p_k \\
1 & p_1 & p_2 & \cdots & p_{k-1} \\
0 & 2 & p_1 & \cdots & p_{k-2} \\
\vdots & \vdots & \vdots & & \vdots \\
0 & 0 & \cdots & \ \ \ 0 \ \ \ k-1 & p_1
\end{array}
\right|.
\]


\medskip
\noin (c) Deduce that if $\lambda_1,\ldots,\lambda_n$ are $n$ complex
numbers such that $p_k(\lambda_1,\ldots,\lambda_n)$ is known for all
$k$ with $1\leq k\leq n$, then $\lambda_1,\ldots,\lambda_n$ are
uniquely determined.

\medskip
\noin
(d) Deduce further that the character $\chi:G\to\C$ of a representation
$V$ of a finite group $G$ determines the eigenvalues of the action
$g:V\to V$ of any element $g\in G$ on $V$.


\skipline
\noin
{\bf B2)} Let $V$ be an irreducible representation of the finite
group $G$. Show that, up to scalars, there is a {\em unique} 
$G$-invariant hermitian inner product on $V$, i.e.\ such that
$\langle gx,gy\rangle = \langle x,y \rangle$ for all $g\in G$ and 
$x,y\in V$. 


\skipline \noin {\bf B3)} Let $G$ be a finite group and let $H$ be a
normal subgroup of $G$ with $|G:H|=n$. Let $V$ be an $n$-dimensional complex
vector space with a basis $b_{g_1H},\ldots, b_{g_nH}$ indexed by the
cosets of $H$. Consider the representation $\rho:G\to \text{GL}(V)$, where
$\rho(g):V\to V$ is defined by $\rho(g)(b_{g_iH}):=b_{gg_iH}$.
Show that $\rho$ is irreducible if and only if $G=H$.

\skipline
\medskip

\begin{center}
{\bf \underline{C problem}}
\end{center}

\skipline
\noin
{\bf C1)}  Suppose that $F$ is a proper subfield of $\C$
such that $\C\supset F$ is a finite extension. The goal of this
problem is to prove that $[\C:F]=2$.

\medskip
\noin
(a) If $[\C:F]=p$ is prime, show that $\C\supset F$ is Galois with
cyclic Galois group, and $F$ contains all $p$-th roots of unity.

\medskip
\noin
(b) If $[\C:F]=p$ is prime and $\sigma$ is a generator of
$G(\C/F)$, show that there exist an $\alpha\in \C\smallsetminus F$,
a $p$-th root $\beta$ of $\alpha$, and a primitive $p^2$-th
root of unity $\gamma$ such that $\alpha^p\in F$ and $\sigma(\beta)=
\gamma\beta$.

\medskip
\noin
(c) Show that if $p$ is an odd prime, then no such $F$ exists.

\medskip
\noin
(d) Show that $[\C:F]=2$, and in fact $\C=F(i)$.

\skipline
\noin
Remark: One can show more generally the following surprising result:
If $K$ is any algebraically closed field which is a proper finite
extension of a subfield $F$, then $K$ must have characteristic zero,
so the above argument applies and $K=F(i)$ for some $i\in K$ such that
$i^2+1=0$.






\end{document}
