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\begin{document}

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


\begin{center}
{\bf PROBLEM SET 11 -- Due May 7, 2026}
\end{center}


\skipline \noin {\bf A1)} Let $V$ be the permutation representation
associated to the action of the finite group $G$ on a finite set $X$, and
$\chi$ be the character of $V$. Let $\chi_1$ denote the trivial
character of $G$.

\medskip
\noin
(a) Show that for $g\in G$, $\chi(g)$ is equal to the number of elements of $X$ that are fixed by $g$.

\medskip
\noin
(b) Show that $\langle\chi, \chi_1\rangle$ is equal to the number of $G$-orbits in $X$.

\skipline
\noin
{\bf A2)} (a) Suppose $G$ is a finite group and 
\[
\dis
G'=[G,G]=\left<xyx^{-1}y^{-1}\ |\ x,y\in G\right> 
\]
is the commutator subgroup of $G$. Prove that the number of
representations of degree 1 of $G$ is equal to the index
$|G:G'|$.

\medskip
\noin
(b) Determine all degree 1 representations of the symmetric group $S_n$.



\skipline
\noin
{\bf A3)} Suppose $G$ is a finite group such that for every normal
subgroup $N$ of $G$, the quotient group $G/N$ is abelian. Let
$\rho : G \to \mathrm{GL}(V)$ be an irreducible representation of 
$G$ with $\dim(V)>1$. Prove that $\rho$ is injective.



\skipline \noin {\bf A4)} Let $V\subset \C[x_1,x_2,x_3]$ be the
$6$-dimensional vector space of homogeneous polynomials in $x_1$, $x_2$,
$x_3$ of degree 2 over $\C$. View $V$ as a representation of $S_3$, with
$S_3$ acting by permuting the variables. Determine the character
$\chi_V$ of the representation $V$ and express $\chi_V$ as a sum of
irreducible characters.


\skipline
\noin
{\bf A5)} Let $\rho:G\to \GL_n(\C)$ be an irreducible representation of a finite group $G$ with character $\chi$, and let $\pi:\GL_n(\C)\to \GL(V)$ be any
complex representation of $\GL_n(\C)$.

\medskip
\noin
(a) Determine the character of the representation $\pi\circ \rho$ of $G$ when $\pi$ is left multiplication of $\GL_n(\C)$ on $V:=M_n(\C)$. Decompose $\pi\circ \rho$ as a direct sum of irreducible representations in this case.

\medskip
\noin
(b) Determine the character of the representation $\pi\circ \rho$ when
$\pi$ is the operation of conjugation on $M_n(\C)$.


\skipline \noin {\bf B1)} If $\rho:G\to \mathrm{GL}(V)$ is a
representation of a finite group $G$, and $\phi:G\to\C$ is any function,
define the {\em Fourier transform} $\wh{\phi}(\rho)$ in $\End(V)$ by
the formula
\[
\dis
\wh{\phi}(\rho)=\sum_{g\in G}\phi(g)\rho(g).
\]

\medskip
\noin
(a) Show that if $\phi$, $\psi$ are any two functions
$G\to \C$, then 
\[
\dis
\wh{\phi *\psi}(\rho)=\wh{\phi}(\rho)\wh{\psi}(\rho).
\]

\medskip
\noin (b) Suppose that $\rho_i:G\to \mathrm{GL}(V_i)$ for $1\leq i\leq
k$ are the irreducible representations of $G$. Prove the {\em Fourier
inversion formula}
\[
\dis
\phi(g)=\frac{1}{|G|}\sum_{i=1}^k(\dim V_i)\cdot \Tr(\rho_i(g^{-1})
\wh{\phi}(\rho_i)).
\]
Deduce that an element $g\in G$ is determined if we know the values
$\rho_i(g)$, that is, if we know the matrices representing $g$ in 
all the irreducible representations of $G$. [Hint: View $\C G$ as a 
direct sum of irreducible representations of $G$, and identify the 
right hand side with the trace of an operator acting on $\C G$.]
 

\medskip
\noin
(c) Prove the {\em Plancherel formula} for functions $\phi$ and $\psi$ 
on $G$:
\[
\dis
\sum_{g\in G} \phi(g)\psi(g^{-1})=\frac{1}{|G|}\sum_{i=1}^k
(\dim V_i)\cdot \Tr(\wh{\phi}(\rho_i)\wh{\psi}(\rho_i)).
\]
[Hint: First take $\phi\leftrightarrow e_g\in \C G$
 to be the characteristic function
of an element $g\in G$ and apply (b).]

\medskip

To see how this problem is related to classical Fourier analysis,
consider the case when $G=S^1=\{z=e^{i\theta}\ |\ \theta\in\R\}$
is the unit circle in the complex plane. A representation of the
compact group $S^1$ is a {\em continuous} group homomorphism $\rho:
S^1\to \mathrm{GL}_n(\C)$. Since $S^1$ is an abelian group, all its
irreducible representations are one dimensional, so they are given by
continuous homomorphisms $S^1\to \C^*$. One can easily show that all
such maps are of the form $\rho_n(z)=z^n$ for $n\in\Z$. In other
words, to each integer $n$ there corresponds a unique irreducible
representation of $S^1$, and vice-versa.

If $f:S^1\to \C$ is a `nice' function (say continuous or at least 
such that $|f|^2$ is integrable -- this latter condition defines the
space $L^2(S^1)$), the Fourier transforms of $f$ as
defined above give a sequence of complex numbers (called the 
`Fourier coefficients' of $f$), namely
\[
\dis
a_n=\wh{f}(n)=\int_{S^1}f(z)z^n\,dz=\frac{1}{2\pi}\int_0^{2\pi}
f(\theta)e^{in\theta}\,d\theta.
\]
(in the second integral, we interpret $f$ as a 
$2\pi$-periodic function on the real $\theta$-line.)
The Fourier inversion formula says that $f$ can be recovered from
its Fourier coefficients:
\[
\dis
f(\theta)=\sum_{n=-\infty}^{+\infty} a_n e^{-in\theta}.
\]
The convergence of the infinite series must be interpreted in the
right way, that is, in the `$L^2$-sense'. In this context the
decomposition of the regular representation
 of $S^1$ into a direct sum of irreducible 
representations is realized as the decomposition of the (infinite
dimensional) {\em Hilbert space} $L^2(S^1)$ given by the orthonormal
basis $\{\rho_n\ |\ n\in\Z\}$. This subject is one of the
beautiful meeting grounds of algebra and analysis.


\skipline
\noin
{\bf B2)} Let $G$ be a finite group and suppose that $V$ is a 
faithful representation of $G$, i.e., $\rho:G\to \mathrm{GL}(V)$ is
injective. Our goal is to prove that any irreducible representation
of $G$ is contained in some tensor power $V^{\otimes n}$ of $V$
(this is a theorem of Burnside and Molien).

\medskip
\noin
(a) If $\chi$ is the character of $V$ and $\psi$ is any irreducible
character, let $a_n=\left<\chi^n,\psi\right>$ and consider the
{\em generating function} $f(t):=\sum a_nt^n$. Show that
\[
\dis
f(t)=\sum_{n=0}^{\infty}a_nt^n=\frac{1}{|G|}
\sum_C\frac{|C|\ov{\psi(C)}}{1-\chi(C)t}.
\]
In the sum the index $C$ runs over all conjugacy classes in $G$.

\medskip
\noin
(b) Show that $\chi(C)=\dim V$ if and only if $C=[1]$.

\medskip
\noin
(c) Deduce that $f(t)$ is a {\em non-constant} rational function, and
finish the proof of the Burnside-Molien theorem.



\skipline \noin {\bf B3)} (a) Let $G$ be a finite group and $g\in
G$. Prove that $g$ is conjugate to $g^{-1}$ if and only if $\chi(g)$
is real for every irreducible character $\chi$ of $G$. We say that the
conjugacy class of $g$ is {\em self-inverse} in this case.

\medskip
\noin (b) Prove that the number of irreducible characters of $G$ that
take only real values (the so-called {\em real characters}) is equal
to the number of self-inverse conjugacy classes of $G$ (the so-called
{\em real classes}).


\skipline
\medskip

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

\skipline \noin {\bf C1)} Let $G$ be a finite group and $\chi$ be the
character of an irreducible representation of $G$. Prove that if
$|\chi(g)|=1$ for some $g\in G$ then $\chi(g)$ is a root of
unity. [Hint: Show that for each $n\in \N$, there are only finitely
  many monic polynomials of degree $n$ in $\Z[x]$ all of whose roots
  have absolute value one. Then use the basic facts about the Galois
  group $G(\Q(\zeta_n)/\Q)$, where $\zeta_n:=e^{2\pi i/n}$, which
  were cited in class.]


\skipline
\noin
{\bf C2)} Let $G$ be a finite group and $\chi$ be the character of an irreducible representation
of $G$ dimension greater than one. Prove that there exists some conjugacy class
$C$ such that $\chi(g)=0$ for all $g\in C$.
[Much easier: prove this under the additional assumption that
$\chi(g)$ is a rational number for all $g\in G$.]



\end{document}
