\documentclass[12pt]{amsart}
\title{Problems in representation theory}
\author{}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb}
\usepackage[all]{xy}

\newcommand{\mattwo}[4]{\left[\begin{array}{cc}
#1 & #2 \\
#3 & #4
\end{array}\right]}

\newcommand{\matthree}[9]{\left[\begin{array}{ccc}
#1 & #2 & #3 \\
#4 & #5 & #6 \\
#7 & #8 & #9 
\end{array}\right]}

\newcommand{\Omit}[1]{}

\newcommand{\g}{\mathfrak{g}}
\newcommand{\Lsl}{\mathfrak{sl}}

\newcommand{\Tr}{\mathrm{Tr}}
%\newcommand{\dim}{\mathrm{dim}}

\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}

\newcommand{\ds}{\displaystyle}

\newcommand{\class}{{\scriptscriptstyle{\mathrm{class}}}}
\newcommand{\Conj}[1]{\mathrm{Conj}(#1)}

\newcommand{\Id}{\mathrm{Id}}
\newcommand{\Image}{\mathrm{Image}}
\newcommand{\Ker}{\mathrm{Ker}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\Ext}{\mathrm{Ext}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\fd}{\scriptscriptstyle{\text{fd}}}
\newcommand{\Aut}{\mathrm{Aut}}

\newcommand{\Ind}{\mathrm{Ind}}
\newcommand{\Res}{\mathrm{Res}}

\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}

\newcommand{\prob}[2]{\noindent {\bf Problem #1. [#2 points]}}

\newcommand{\Sym}{\mathrm{Sym}}

\begin{document}
\maketitle

\hrule
\begin{center}
{\em In Problems 1--10, $k$ is a field and
$A$ is a unital, associative algebra over $k$.}
\end{center}
\hrule

\vspace*{0.1in}

\prob{1}{10} Let $V$ be a representation of $A$.
\begin{itemize}
\item[(a)] Prove that $V$ is irreducible if, and only if, for every
non--zero $v\in V$, we have: $A\cdot v = V$. Here, $A\cdot v :=
\{a\cdot v : a\in A\}$.
\item[(b)] We say that $V$ is co--cyclic if there exists
$0\neq v\in V$ such that $v\in U$ for every non--zero subrepresentation
$U\subset V$. Prove that every co--cyclic representation is
indecomposable.
\item[(c)] Let $A = \C[x,y]/(x^2,xy,y^2)$. Prove that $A$ is indecomposable
(as $A$--representation) and $A$ is not co--cyclic. Consider $\rho:A\to\End_\C(\C^3)$
given by:
\[
\rho(x) = \matthree{0}{0}{0}{0}{0}{0}{1}{0}{0},\qquad
\rho(y) = \matthree{0}{0}{0}{0}{0}{0}{0}{1}{0}.
\]
Show that $\rho$ a co--cyclic (hence indecomposable) representation of $A$.\\
\end{itemize}

\prob{2}{15} Let $K$ be an extension of $k$ (i.e., $k$ is a subfield of $K$), 
and let $f_1,\ldots, f_m : A\to K$
be $m$ distinct homomorphisms of $k$--algebras. Prove that $f_1,\ldots, f_m$
are linearly independent over $K$.\\

\prob{3}{10} Assume that $k$ is algebraically closed in this problem.
Let $V$ be a finite--dimensional representation of $A$.
Prove that $V$ is semisimple if, and only if, for every subrepresentation
$U\subset V$, there exists a subrepresentation $W\subset V$ such that
$V\cong U\oplus W$.\\

\prob{4}{10} Show that the following representations are irreducible.
\begin{itemize}
\item[(a)] $A = \mathrm{Mat}_{n\times n}(k)$, $V = k^n$.
\item[(b)] $A = \C\left\langle X,D\right\rangle/(DX-XD-1)$, $V = \C[x]$
with the action given by $X\cdot f(x) = xf(x)$ and
$D\cdot f(x) = f'(x)$
(derivative of $f$).\\
\end{itemize}

\prob{5}{10} (Idempotent decomposition) Assume that there exist $e_1,\ldots,e_n\in A$ such that
\[
1 = e_1+\cdots +e_n,\qquad e_ie_j = \delta_{ij} e_i,\ \forall\ 1\leq i,j\leq n.
\]
Prove that for any $A$--representation $V$, we have a direct sum
decomposition of vector spaces: $V \cong \oplus_{i=1}^n e_i\cdot V$.\\

\prob{6}{10} (Tensor-Hom adjointness) Let $V$ and $W$ be two vector
spaces over $k$. Write the natural map $\alpha_{V,W} : W\otimes V^*
\to \Hom_k(V,W)$. Prove that this map is an isomorphism, assuming
$V$ is finite--dimensional. Here, $V^* = \Hom_k(V,k)$ is the dual
vector space.\\

\prob{7}{10} Assume that $k$ is algebraically closed.
Let $V$ be a finite--dimensional, irreducible representation
of $A$. Let $D_1$ and $D_2$ be two finite--dimensional vector spaces over
$k$. Recall that we have an isomorphism:
\[
\Hom_k(D_1,D_2) \stackrel{\scriptstyle{\varphi}}{\rightarrow}
\Hom_A(V\otimes_k D_1, V\otimes_k D_2)
\] 
given by: $\varphi(X)(v\otimes \xi) = v\otimes X(\xi)$, for every $X\in\Hom_k(D_1,D_2)$,
$v\in V$ and $\xi\in D_1$.
Prove that (a) $\Ker(\varphi(X)) = V\otimes \Ker(X)$ and (b)
$\Image(\varphi(X)) = V\otimes \Image(X)$.\\

\prob{8}{10} Let $V,W$ be two finite--dimensional representations of $A$.
Assume that there exists a finite extension $L$ of $k$ such that
$V\otimes_k L$ is isomorphic to $W\otimes_kL$, as representations of
$A\otimes_kL$. Prove that $V$ and $W$ are isomorphic (as representations of
$A$). [Problem 3.8.4 of Pasha's book.]\\

\prob{9}{15} Let $A = \{f:\R\to\R \text{ continuous } | f(x+1)=f(x)\}$
and let $M = \{g:\R\to\R \text{ continuous } | g(x+1)=-g(x)\}$. Prove that
$A$ and $M$ are non--isomorphic, indecomposable representations of $A$.
Show that $A\oplus A$ is isomorphic to $M\oplus M$.\\

\prob{10}{10} (See Problem 3.9.1 of Pasha's book). Let $V,W$
be two representations of $A$. Given a linear map $f:A\to \Hom_k(W,V)$,
define $\rho_f : A \to \End_k(V\oplus W)$ given by:
\[
\rho_f(a) = \mattwo{\rho_1(a)}{f(a)}{0}{\rho_2(a)}.
\]
\begin{itemize}
\item[(a)] Show that $\rho_f$ is a representation of $A$ if, and only
if $f(ab) = \rho_1(a)f(b) + f(a)\rho_2(b)$, for every $a,b\in A$.
Let the space of all the solutions of this equation be denoted by
$Z^1(W,V)\subset \Hom_k(A,\Hom_k(W,V))$ (called $1$--cocycles).

\item[(b)] Given $X\in\Hom_k(W,V)$, define the automorphism of
the vector space $g\in \Aut_k(V\oplus W)$ given by:
\[
g = \mattwo{\Id_{V_1}}{X}{0}{\Id_{V_2}}.
\]
Let $f:A\to\Hom_k(W,V)$ be an element of $Z^1(W,V)$ as in part (a).
Show that $g$ is an isomorphism between $\rho_f$ and $\rho_0$
if, and only if $f(a) = \rho_1(a)X - X\rho_2(a)$ for every
$a\in A$. Given $X\in\Hom_k(W,V)$, let $\delta(X) : A\to \Hom_k(W,V)$
be defined as: $\delta(X)(a) = \rho_1(a)X-X\rho_2(a)$.\\
\end{itemize}

\prob{10 cntd.}{20} (same notations as in Problem 10 above).
Let
$B^1(W,V) = \{\delta(X) : X\in\Hom_k(W,V)\}$. Verify that
$B^1(W,V) \subset Z^1(W,V)$. Define $\Ext_A^1(W,V) :=
Z^1(W,V)/B^1(W,V)$.

Let $0\to V_1\to V_2\to V_3\to 0$ be a short
exact sequence of $A$--linear maps between $A$--representations.
Show that, for every representation $W$ of $A$,
 we get the following long exact sequence of vector
spaces (describe the connecting homomorphism $\delta$ explicitly):
\[
\xy
(-3,20)*{0}="i";
(20,20)*{\Hom_A(W,V_1)}="rfx"; {\ar "i"; "rfx"};
(50,20)*{\Hom_A(W,V_2)}="rfy"; {\ar "rfx"; "rfy"};
(80,20)*{\Hom_A(W,V_3)}="rfz"; {\ar "rfy"; "rfz"};
{\ar "rfz"; (100,20)};
(100,20); (100,10) **\crv{(110,20)&(110,10)};
{\ar@{-} (100,10); (0,10)}; (50,12)*{\scriptstyle{\delta}};
(0,10); (0,0) **\crv{(-10,10)&(-10,0)};
(20,0)*{\Ext^1(W,V_1)}="rrfx"; {\ar (0,0); "rrfx"};
(50,0)*{\Ext^1(W,V_2)}="rrfy"; {\ar "rrfx"; "rrfy"};
(80,0)*{\Ext^1(W,V_3)}="rrfz"; {\ar "rrfy"; "rrfz"};
%(95,0)*{\cdots}
\endxy
\]

\hrule\hrule

\prob{11}{15} Let $G$ be a finite group acting on a finite set $X$ via
group homomorphism $\alpha:G\to S_X$ (here, $S_X$ is the group of permutations
of elements of $X$). Let $V$ be the vector space (over $\C$) of all
complex--valued functions on $X$. We view $V$ as a $G$--representation,
$\rho:G\to\GL(V)$, by:
\[
(\rho(g)(f))(x) := f(\alpha(g)^{-1}(x)),\ \text{for every } g\in G, f\in V, x\in X.
\]

\begin{enumerate}
\item Prove that $\dim(V^G) = |X/G|$ (number of $G$--orbits in $X$).
\item For $g\in G$, show that $\Tr(\rho(g)) = |X^g|$. Here, $X^g := \{x\in X : 
\alpha(g)(x) = x\}$.
\item Consider the averaging operator
$\displaystyle P = \frac{1}{|G|}\sum_{g\in G} \rho(g)$, acting on $V$. Show that
$\Tr(P) = \dim(V^G)$. Hence we obtain {\em Burnside's orbit counting formula}
\[
|X/G| = \frac{1}{|G|} \sum_{g\in G} |X^g|.
\]
\end{enumerate}

\prob{12}{10} Let $G$ be a finite group and $V$ a finite--dimensional $G$--representation
over $\C$. Assume that $\chi_V(g) = 0$ for every $g\in G\setminus\{e\}$. Prove
that $\dim(V)$ is divisible by $|G|$.\\

\prob{13}{20} Let $D_n$ be the dihedral group (symmetries of regular $n$--gon, thus
$|D_n|=2n$). In the problems below, the representations of $D_n$ are over $\C$.
\begin{enumerate}
\item Count the number of conjugacy classes in $D_n$.
\item Show that $D_n$ has exactly two $1$--dimensional representations, if $n$
is odd; and exactly four $1$--dimensional representations, if $n$ is even.
\item Let $V$ be an irreducible representation of $D_n$. Show that $\dim(V)\leq 2$.
\item Give explicit description of all irreducible $D_n$--representations.\\
\end{enumerate}

\prob{14}{10} Let $S_n$ be the group of permutations of $n$ letters. Let
$\C^n$ be the standard $n$--dimensional $S_n$--representation (i.e., in
a basis $e_1,\ldots,e_n$ of $\C^n$: $\sigma\cdot e_i = e_{\sigma(i)}$,
for every $\sigma\in S_n$ and $1\leq i\leq n$). Show that $\C^n$ is
a direct sum of two irreducible $S_n$--representations. Give a basis
of each of these subrepresentations of $\C^n$.\\

\noindent {\em Notations for Problems 15 and 16:} Let $G$ be a finite group, $\C[G]$
its group algebra over $\C$. Recall the convolution product, for $f_1,f_2\in\C[G]$:
\[
(f_1\ast f_2)(g) = \sum_{\sigma\in G} f_1(\sigma)f_2(\sigma^{-1}g).
\]
We defined a symmetric, bilinear form $\beta:\C[G]\times\C[G]\to\C$ by:
\[
\beta(f_1,f_2) = \frac{1}{|G|} (f_1\ast f_2)(e) = \frac{1}{|G|}
\sum_{g\in G} f_1(g^{-1})f_2(g)\ .
\]
Let $\C[G]_\class$ denote the (commutative) subalgebra of functions
constant on conjugacy classes of $G$ (abbreviated to $\Conj(G)$).\\

Let $\{V_\lambda:\lambda\in P(G)\}$ be the set of isomorphism classes
of finite--dimensional, irreducible $G$--representations. Let $\rho_\lambda$
denote the group homomorphism $G\to \GL(V_\lambda)$, or the corresponding
algebra homomorphism $\C[G]\to \End(V_\lambda)$. Also, let $\chi_\lambda$
denote the character of the representation $V_\lambda$.\\

\prob{15}{10} Write the change of bases formulae for the following three
bases for $\C[G]_{\class}$: (i) $\{\delta_C : C\in\Conj{G}\}$,
(ii) $\{\chi_\lambda : \lambda\in P(G)\}$, and (iii)
$\{P_\lambda : \lambda\in P(G)\}$, where
\[
P_\lambda := \frac{\dim(V_\lambda)}{|G|} \sum_{g\in G} \chi_\lambda(g^{-1}) \delta_g
\in \C[G]_\class.
\]

\prob{16}{15} {\em Plancheral formula.}
For $f_1,f_2\in\C[G]$, show that:
\[
\beta(f_1,f_2) = \frac{1}{|G|^2} \sum_{\lambda\in P(G)}
\dim(V_\lambda) \Tr(\rho_\lambda(f_1\ast f_2)).
\]

\prob{17}{15} Consider the following isomorphisms:
\[
\psi:\C[G]\to\bigoplus_{\lambda\in P(G)} \End(V_\lambda),\qquad
\varphi : \bigoplus_{\lambda\in P(G)} V_\lambda\otimes V_{\lambda}^*
\to 
\bigoplus_{\lambda\in P(G)} \End(V_\lambda).
\]
\begin{enumerate}
\item Show that $\psi$ and $\varphi$ are $G\times G$--intertwiners. 
\item For $v\in V_\lambda$ and $\xi\in V_\lambda^*$, consider the
following function on $G$ (called {\em matrix coefficient}):
$c_{v,\xi}(g) = \xi(g^{-1}\cdot v)$.\\

Show that $\displaystyle \psi^{-1}(\varphi(v\otimes \xi)) = \frac{\dim(V_\lambda)}{|G|}
c_{v,\xi}$.
\end{enumerate}

\prob{18}{10} Let $G,H$ be two finite groups. Prove that irreducible,
finite--dimensional $G\times H$--representations are precisely
$V\otimes W$, where $V$ and $W$ are irreducible, finite--dimensional 
representations of $G$ and $H$ respectively.\\

\prob{19}{20} Compute the character table of $S_5$.\\

\hrule\hrule

\vspace*{0.2in}

\prob{20}{15} Let $\ds f(t) = \sum_{n=0}^{\infty} f_n \frac{t^n}{n!}$
and $\ds g(t)=\sum_{\ell=1}^{\infty} g_\ell t^\ell$ be two formal power
series with coefficients from an arbitrary commutative algebra over
$\Q$. Note that $g(0)=0$. Prove that:
\[
f(g(t)) = \sum_{n=0}^{\infty} f_n \frac{g(t)^n}{n!} = 
f_0 + \sum_{N=1}^{\infty} t^N \lp \sum_{k=1}^N
f_k \lp \sum_{\begin{subarray}{c} \lambda\vdash N \\ \ell(\lambda)=k
\end{subarray}} \frac{g_\lambda}{\prod_{i\geq 1} \ell_i!} \rp \rp\ ,
\]
where $\ell_i = |\{r : \lambda_r=i\}|$ and $g_\lambda = g_{\lambda_1}\cdots
g_{\lambda_k}$.\\


\prob{21}{15} Recall the definition of elementary, complete and power
sum symmetric polynomials, in $N$ variable. Convention: $e_0 = h_0 = 1$.
\[
e_r := \sum_{1\leq i_1<\ldots <i_r\leq N} x_{i_1}\cdots x_{i_r},
\qquad
p_r := \sum_{i=1}^N x_i^r,
\qquad
h_r := \sum_{1\leq i_1\leq\ldots \leq i_r\leq N} x_{i_1}\cdots x_{i_r}.
\]

Consider the generating series:
\[
E(t) = \sum_{r=0}^N e_r t^r,\qquad
H(t) = \sum_{r=0}^{\infty} h_r t^r,\qquad
P(t) = \sum_{n=1}^{\infty} p_n \frac{t^n}{n}.
\]

\begin{itemize}
\item[(a)] Show that $H(t)E(-t)=1$ and $H(t) = \exp\lp P(t)\rp$.
\item[(b)] Prove {\em Newton's identities}:
\[
nh_n = \sum_{r=1}^n p_r h_{n-r},\qquad
ne_n = \sum_{r=1}^n (-1)^{r-1} p_r e_{n-r}.
\]
\end{itemize}

\prob{22}{10} Let $n\in\Z_{\geq 2}$ and $\lambda\vdash n$. Let
$V_\lambda$ be the finite--dimensional, irreducible representation
of the symmetric group $S_n$, labelled by $\lambda$, as in Frobenius'
theorem. Let $w = (1\ 2\ \ldots\ n)\in S_n$. Prove that
\[
\operatorname{Trace}(w \text{ acting on } V_\lambda) = 
\left\{
\begin{array}{ll}
(-1)^k,&\text{if } \lambda=(n-k,1,\ldots,1), \\
0 &\text{otherwise}.
\end{array}
\right.
\]

\prob{23}{10} Let $\lambda,\mu\vdash n$. Recall the notation:
\[
X_\lambda = \left\{ (I_1,\ldots,I_\ell) : 
\{1,\ldots,n\} = \bigsqcup_{i=1}^\ell I_i \text{ and } 
|I_j| = \lambda_j, \forall j
\right\}
\]
\begin{itemize}
\item[(a)] Show that $X_\lambda$ is in bijection with
$S_n/S_\lambda$, where $S_\lambda = S_{\lambda_1}\times \cdots S_{\lambda_\ell}
< S_n$.
\item[(b)] Show that there is a bijection between the following
three sets:
\begin{enumerate}
\item $(X_\lambda\times X_\mu)/S_n$, 
\item $X_\mu^{S_\lambda} :=
\{a\in X_\mu : w\cdot a = a,\ \forall w\in S_\lambda\}$,
\item $\{A = (a_{ij}) \in\mathrm{M}_{n\times n}(\Z_{\geq 0}):
\sum_j a_{ij} = \lambda_i, \forall i \text{ and }
\sum_i a_{ij} = \mu_j, \forall j\}$. \\
\end{enumerate}
\end{itemize}


\prob{24}{15} Recall the definition of $(\cdot, \cdot)$ on 
$\Lambda_n = \Z[x_1,\ldots, x_N]^{S_N}_{\text{deg}=n}$.
\begin{enumerate}
\item For $\lambda\vdash n$, let $h_\lambda = h_{\lambda_1}\cdots h_{\lambda_\ell}$
and $e_{\lambda} = e_{\lambda_1}\cdots e_{\lambda_\ell}$, where $h_r$ and $e_r$
are defined above (Problem 21). Show that $(h_\lambda, e_\mu)$ is
the number of matrices with $0,1$ entries whose row sums equal $\lambda$
and column sums equal $\mu$.
\[
(h_\lambda, e_\mu) = \left|\left\{ A = (a_{ij}) : 
a_{ij}\in\{0,1\}\ \forall i,j \text{ and } \sum_{j} a_{ij} = \lambda_i,\ \forall i;
\sum_i a_{ij} = \mu_j,\ \forall j \right\}\right|
\]
\item Let $s_\lambda$ be the Schur polynomial, and $m_\mu$ be the monomial
symmetric polynomial. Let $\rho = (N-1, N-2, \ldots, 0)$. Show that:
\[
\boxed{
(s_\lambda, m_\mu) = \sum_{\begin{subarray}{c}
w\in S_N : \\ \lambda + \rho - w(\rho) \in S_N\cdot\mu
\end{subarray}} \varepsilon(w)
}
\]
\end{enumerate}

\prob{25}{10} Let $\mu\vdash (n-1)$. Prove the following ({\em Pieri's rule}):
\[
s_\mu s_{(1)} = \sum s_\lambda\ ,
\]
where the sum is over all $\lambda\vdash n$ which can be obtained from
$\mu$ by adding exactly one box (i.e., $\exists j:$ $\lambda_i=\mu_i$,
$\forall i\neq j,$ and $\lambda_j = \mu_j+1$).\\

\prob{26}{10} Let $G$ be a finite group, $H<G$ a subgroup and let
${\mathbf 1}$ denote the one dimensional, trivial representation of $H$.
Prove that:
\[
\chi_{\Ind_H^G{{\mathbf 1}}}(g) = |(G/H)^g| = \frac{|C\cap H|\cdot |Z_G(g)|}{|H|}\ ,
\]
where $C = \{xgx^{-1} : x\in G\}$ is the conjugacy class containing $g$,
and $Z_G(g) = \{x\in G : xg=gx\}$ is its centralizer.\\

\prob{27}{10} Let $\lambda\vdash n$ and let $\iota_\lambda$
is the character of the induced representation $U_\lambda$ of $S_n$.
Let $\mu\vdash n$ be another partition and let $r_i=|\{j:\mu_j=i\}|$.
Use the previous exercise to prove the following formula:
\[
\iota_\lambda(\mu) = \sum_{(s_{ij})} \prod_{i\geq 1} \frac{r_i!}{\prod_j s_{ij}!}\ ,
\]
where the sum is over all matrices with $\Z_{\geq 0}$ entries such that
$\sum_j s_{ij} = r_i$ and $\sum_i is_{ij} = \lambda_i$.\\


\prob{28}{10} (Reciprocity formula) Let $G$ be a finite group and 
$H<G$ a subgroup. Let $\phi\in \C[G]_\class$ and
$\psi\in \C[H]_\class$. Prove that
\[
\lp \Res_H^G\phi, \psi\rp_H = \lp \phi, \Ind_H^G\psi \rp_G\ .
\]

\hrule\hrule

\vspace*{0.2in}

\prob{29}{10} Let $m\in\Z_{\geq 2}$. Prove directly that, for any $r\in\Z_{\geq 0}$,
$\Sym^r(\C^m)$ is an irreducible representation on $\GL_m(\C)$.\\

\prob{30}{10} Let $\{R_1,\ldots, R_N\}$ be finite--dimensional
representations of a group $G$. Show that these are pairwise non-isomorphic,
irreducible representations of $G$ if, and only if,
the matrix coefficients of $R_1,\ldots,R_N$ are linearly independent.\\

\prob{31}{15} Recall that, for a graded vector space
$U = \oplus_{n=0}^{\infty} U[n]$, such that $\dim(U[n])<\infty$,
for every $n\geq 0$, we define:
\[
\text{(Hilbert polynomial)}\qquad P(U;t) = \sum_{n\geq 0}
\dim(U[n]) t^n.
\]

Let $V$ be a finite-dimensional vector space, and let
$T^{\bullet}(V) = \oplus_{n\geq 0} T^n(V)$, and similarly
$\Sym^{\bullet}(V), \wedge^{\bullet}(V)$. Prove that:
\[
P(T^{\bullet}(V); t) = \frac{1}{1-\dim(V)t}\ ,\qquad
P(\Sym^{\bullet}(V); t) = \frac{1}{(1-t)^{\dim(V)}}\ ,
\]
\[
P(\wedge^{\bullet}(V); t) = (1+t)^{\dim(V)}\ .
\]

\prob{32}{10} Let $\lambda=(\lambda_1\geq \ldots \geq \lambda_\ell)$
be a partition, and let $m\geq\ell$. Show that we have an isomorphism
of $\GL_m(\C)$--representations:
\[
L_{\lambda+(1^\ell)} \cong L_\lambda\otimes\wedge^\ell(\C^m).
\]

\prob{33}{15} {\em Branching rule for general linear group.}
Let $\lambda$ be a partition of length $\ell$, and $m\geq \ell$.
Let $L^{(m)}_\lambda$ denote the corresponding representation
of $\GL_m(\C)$. Consider $\GL_{m-1}(\C)$ as the subgroup
of $\GL_m(\C)$ via $g\mapsto \mattwo{g}{0}{0}{1}$. Show that:
\[
\left.L^{(m)}_\lambda\right|_{\GL_{m-1}(\C)}
\cong
\bigoplus_\mu L_\mu^{(m-1)}\ ,
\]
where the direct sum is over all $\mu=(\mu_1\geq\ldots\geq\mu_\ell\geq 0)$
satisfying the interlacing condition:
\[
\lambda_1\geq \mu_1\geq \lambda_2\geq \mu_2 \geq
\cdots \geq \lambda_\ell\geq\mu_\ell\ .
\]
In other words, the Young diagram of $\mu$ is contained in
the Young diagram of $\lambda$ and its complement consists
entirely of horizontal strips.\\

\prob{34}{10} Let $m\geq\Z_{\geq 2}$ and assume that
$\{d_n\}_{n=1}^{\infty} \subset\Z_{\geq 0}$ are defined by
the following identity:
\[
(1-mt) = \prod_{n=1}^{\infty} (1-t^n)^{d_n}\ .
\]
Show that $\ds d_n = \frac{1}{n} \sum_{d|n} \mu(d) m^{\frac{n}{d}}$.
Here $\mu:\Z_{\geq 1} \to \{0,\pm 1\}$ is the M\"{o}bius
function defined as: $\mu(1)=1$, $\mu(r)=0$ if there is a prime
$p$ such that $p^2$ divides $r$, and for distinct primes
$p_1,\ldots,p_k$, we have $\mu(p_1\cdots p_k) = (-1)^k$.\\

\hrule\hrule

\vspace*{0.2in}

{\em Notations:} The following problems are about representation
theory of $\Lsl_2$ (over $\C$). Recall that for every $n\in\Z_{\geq 0}$,
$L_n$ denotes the irreducible $n+1$--dimensional representation
of $\Lsl_2$ described explicitly as follows.
$L_n$ has a basis $\{v_0,\ldots,v_n\}$ such that:
\[
h\cdot v_\ell = (n-2\ell)v_\ell,\qquad
e\cdot v_\ell = (n-\ell+1)v_{\ell-1},\qquad
f\cdot v_\ell = (\ell+1)v_{\ell+1},
\]
with the understanding that $v_{-1}=v_{n+1}=0$.\\

\prob{39}{10} Prove that $L_n\otimes L_1 \cong L_{n+1}\oplus L_{n-1}$
as $\Lsl_2$--representations.\\

\prob{40}{50} Exercise 2.15.1 of Pasha's book.\\

\prob{41}{30} Exercise 2.16.4 of Pasha's book. Classify all
finite--dimensional, irreducible representations of $\Lsl_2$
over an algebraically closed field of characteristic $p>2$.\\

\prob{42}{30} Exercise 2.16.5 of Pasha's book.\\

\prob{43}{20} Note that the action of $e$ and $f$ on $L_n$
is nilpotent. For any nilpotent operator $X$, the following
is a well--defined invertible operator:
\[
\exp(X) = 1 + X + \frac{X^2}{2!} + \ldots
\]
Let $\mathsf{s} := \exp(e)\exp(-f)\exp(e)$, considered as a
operator on $L_n$. Show that:
\[
\mathsf{s}\cdot v_\ell = (-1)^{n-\ell} v_{n-\ell}.
\]

\prob{44}{20} For any $\lambda\in\C$, define $M_\lambda$
to be (infinite--dimensional) $\Lsl_2$--representation as
follows: $M_\lambda$ has a basis $\{m_r:r\in\Z_{\geq 0}\}$
with $\Lsl_2$--action given by (with the understanding
that $m_{-1}=0$):
\[
h\cdot m_\ell = (\lambda-2\ell)m_\ell,\qquad
e\cdot m_\ell = (\lambda-\ell+1)m_{\ell-1},\qquad
f\cdot m_\ell = (\ell+1)m_{\ell+1}.
\]
Show that $M_\lambda$ is irreducible for $\lambda\not\in\Z_{\geq 0}$.
For $\lambda\in\Z_{\geq 0}$, show that $M_\lambda$
has a subrepresentation isomorphic to $M_{-\lambda-2}$,
so that we have the following non--split short exact
sequence of $\Lsl_2$--representations:
\[
0\to M_{-\lambda-2} \to M_\lambda \to L_\lambda \to 0\ .
\]

\vspace*{0.1in}

\begin{center}
{\em Problems 45 and 46 are inspired by the following beautiful
paper by Etingof and Varchenko: Dynamical Weyl group
and applications, Advances in Mathematics,} {\bf 167} 
{\em (2002) pp. 74--127}.\\
\end{center}

\vspace*{0.1in}

\prob{45}{30} Let $\lambda\in\C$ and $n\in\Z_{\geq 0}$.
Let $M_\lambda$ be the Verma module (see Problem 44) with
basis denoted by $\{m^{(\lambda)}_r : r\in\Z_{\geq 0}\}$.
Let $L_n$ be the $n+1$--dimensional irreducible representation
of $\Lsl_2$, with basis denoted by $\{v_0,\ldots,v_n\}$. 
Let $0\leq k\leq n$.\\

Show that, for all but finitely many $\lambda\in\C$,
there is a unique $\Lsl_2$--intertwiner:
\[
\Psi^{(n,k)}_\lambda : M_\lambda \to M_\mu\otimes L_n\ \,
\]
where $\mu = \lambda - (n-2k)$, whose value on $m^{(\lambda)}_0$
is of the following form, for uniquely determined $c_r\in\C$:
\[
\Psi^{(n,k)}_\lambda : m^{(\lambda)}_0 \mapsto
m^{(\mu)}_0\otimes v_k + \sum_{r=1}^{k} c_r m^{(\mu)}_r\otimes v_{k-r}\ .
\]

Deduce that we have the following isomorphism (generically in
$\lambda$):
\[
\Hom_{\Lsl_2}(M_\lambda, M_\mu\otimes V) \to
V[\lambda-\mu]
\]
sending an intertwiner $\psi$ to the coefficient of $m^{(\mu)}_0$
in $\psi(m^{(\lambda)}_0)$.\\

\prob{46}{30} (continued from Problems 44, 45). Now assume
that $\lambda\in\Z_{\geq 0}$ and $\lambda\gg 0$. Let $n, k$
and $\Psi^{(n,k)}_\lambda$ be as in Problem 45 above.
Show that:
\[
\Psi^{(n,k)}_\lambda \lp m^{(\lambda)}_{\lambda+1} \rp
 = m^{(\mu)}_{\mu+1} \otimes (a(\lambda) v_{n-k}) + 
\sum_{r\geq 1} b_r(\lambda) m^{(\mu)}_{\mu+1+r} \otimes v_{n-k-r}
\]
for some $a(\lambda), b_r(\lambda)$. Compute $a(\lambda)$
explicitly.\\

\hrule\hrule

\vspace*{0.2in}

\Omit{

\prob{47}{15} Let $A$ be a symmetric $n\times n$ matrix with real entries.
Prove that the following are equivalent.
\begin{itemize}
\item $A$ is positive--definite. That is, for every $v\in \R^n$,
$v^TAv \geq 0$, with equality if, and only if $v=0$.
\item All the principal minors of $A$ are positive. That is,
for every $1\leq i_1<\ldots <i_\ell\leq n$, we have:
\[
\mathrm{det}\lp a_{i_j,i_k}\rp_{1\leq j,k\leq \ell} > 0.
\]
\item There exists $u\in\R^n_{>0}$ such that $Au \in \R_{>0}^n$.
\end{itemize}

{\em Notations.} Let $\bfI$ be a finite indexing set, $\bfA=(a_{ij})_{i,j\in\bfI}$
be a Cartan matrix, $\{d_i:i\in\bfI\}$ the symmetrizing integers.
$E$ is the real vector space with basis $\{h_i:i\in\bfI\}$.
$\{\alpha_j\}_{j\in\bfI}\subset E^*$ is the basis defined
by $\alpha_j(h_i) = a_{ij},\ \forall i,j\in\bfI$.
We have the following positive--definite forms on $E$
and $E^*$:
\[
(\alpha_i,\alpha_j) = d_ia_{ij};\qquad
(h_i,h_j) = a_{ij}d_j^{-1}.
\]
Let $s_i$ denote the orthogonal reflection defined by:
\[
s_i(\gamma) = \gamma - \gamma(h_i)\alpha_i; \qquad
s_i(h) = h - \alpha_i(h) h_i\ .
\]
The Weyl group $W$ is the group generated by $\{s_i\}_{i\in\bfI}$.
The identification $\phi:E^*\to E$, given by $\phi(\alpha_i) = d_ih_i$,
is an isometry and commutes with $W$--action.
$R\subset E^*$ is the set of roots. Recall: $R = R_+\sqcup R_-$.\\


\prob{48}{10} Prove that for every $w\in W$, the length
$\ell(w) = |\{\alpha\in R_+ : w(\alpha)\in R_-\}|$.\\

}

\end{document}
