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\begin{document}

\title[]{MATH 7382: TOPICS IN PROBABILITY\\ Problem Set 2}
\author[]{Due: 10/16/12}
\maketitle



(* means harder optional exercises)

\medskip


\textbf{1.}
(a) Show that the finite-dimensional unitary group\footnote{Which consists of all complex $N\times N$ matrices $U$ with condition $UU^*=E$; here $E$ is the identity matrix and $U^*$ means transposed complex conjugate matrix.}  $U(N)$ is compact in the usual topology. 

(b) Show that the infinite-dimensional unitary group $U(\infty)=\bigcup_{N=1}^{\infty}U(N)$ with the inductive limit topology (= natural topology of convergence of all matrix elements) is not compact. 

(c)* Show that $U(\infty)$ is not locally compact.

\medskip

\textbf{2.}
In the algebra of Laurent polynomials
$\C\left[ u_1^{\pm1},\dots,u_N^{\pm1} \right]$
(as well as in the subalgebra of symmetric polynomials $\LSym(N)$) consider the automorphism
\begin{equation*}
u_i\mapsto u_i^{-1},\qquad \mbox{for all }i=1,\dots,N.
\end{equation*}
Show that $s_\la(u^{-1})=s_{\la^*}(u)$, where
\begin{equation*}
\la^*=(-\la_N,-\la_{N-1},\dots,-\la_2,-\la_1).
\end{equation*}

\medskip

\textbf{3.} [elementary and complete homogeneous symmetric polynomials] Let $e_k(u_1,\ldots,u_N)$, $k=1,\ldots,N$, be the $k$th elementary symmetric polynomial: $e_k(u_1,\ldots,u_N)=\sum\limits_{1\le i_1<\ldots< i_k\le N} u_{i_1}\ldots u_{i_k}$. Also, let $h_k(u_1,\ldots,u_N)$, $k=1,2,\ldots$, be the $k$th complete homogeneous symmetric polynomial: $h_k(u_1,\ldots,u_N)=\sum\limits_{1\le i_1\le\ldots\le i_k\le N} u_{i_1}\ldots u_{i_k}$.

(a) Show that $e_k(u_1,\ldots,u_N)=s_{(\underbrace{\scriptstyle1,\ldots,1}_{k},0,\ldots,0)}(u_1,\ldots,u_N)$; 

(b) Show that $h_k(u_1,\ldots,u_N)=s_{(k,0,\ldots,0)}(u_1,\ldots,u_N)$.

\textbf{Hint:} You may use the branching rule for the Schur polynomials in one of the two ways. (1) Write $e_k(u_1,\ldots,u_{N-1},1)$ in terms of $e_j$'s in variables $(u_1,\ldots,u_{N-1})$; and same for $h_k$'s. Compare this with the branching rule for Schur polynomials. (2) Use combinatorial formula for Schur polynomials directly.

\medskip

\textbf{4.} [Schur polynomials in two variables; cf. Problem 2 in the first Problem Set] Let $N=2$, and consider the Schur polynomials in two variables:
\begin{align*}
	s_{(a,b)}(x,y)=
	\frac{x^{a+1}y^{b}-x^{b}y^{a+1}}{x-y},\quad a\ge b\ge0.
\end{align*}
Write this Schur polynomial as a linear combination of the symmetrized monomials
\begin{align*}
	m_{(c,d)}(x,y):=\begin{cases}
	x^{c}y^{d}+x^{d}y^{c},& c>d\ge0;\\
	x^{c}y^{c},& c=d\ge0.
	\end{cases}
\end{align*}
\textbf{Explanation:} In this way you get the Kostka numbers $K_{\la\mu}$ which are defined as 
\begin{align*}
	s_\la=\sum_{\mu}K_{\la\mu}m_{\mu}
\end{align*}
for all $\la$'s and $\mu$'s with two parts.


\medskip

\textbf{5.} Show that the number $\dim\la$ of standard Young tableaux of shape $\la$ ($\la$ is a nonnegative signature)\footnote{Recall also that $\dim\la$ is the dimension of the irreducible representation of $\SG(|\la|)$ corresponding to $\la$.} is equal to the coefficient by $x_1^{\la_1+N-1}x_2^{\la_2+N-2}\ldots x_N^{\la_N}$ in the polynomial
\begin{align*}
	(x_1+\ldots+x_N)^{N}\prod_{1\le i<j\le N}(x_i-x_j),
\end{align*}
where $N$ is large enough ($N\ge \ell(\la)$).

\textbf{Hint:} use the corollary of the Pieri formula: $p_1^{K}=\sum_{\la\in\GT^+
\colon|\la|=K}\dim\la\cdot s_\la$, and multiply by the Vandermonde.

\medskip

\textbf{6.} [Plancherel specialization of the algebra of symmetric functions] Let us consider a specialization of the algebra of symmetric functions $\Sym=\R[p_1,p_2,\ldots]$ (i.e. algebra morphism $\Sym\to\R$) defined on the multiplicative generators (power sums) $p_k$ as 
\begin{align*}
	p_1\to1,\qquad p_2,p_3,p_4,\ldots \to0.
\end{align*}

(a) Show that under this specialization $e_k$ and $h_k$ map to ${1}/{k!}$. \textbf{Hint:} use relations between the generating functions of $p$'s, $e$'s and $h$'s.

(b) Show that $s_{\la}$ maps to $\displaystyle\det\left[\frac{1}{(\la_i-i+j)!}\right]_{i,j=1}^{N}$ (for any $N\ge \ell(\la)$). \textbf{Hint:} use Jacobi-Trudi identity.

(c)* In fact, the image of $s_\la$ is equal to $\dim\la/|\la|!$.

\medskip

\textbf{7.}
Let $\la$ be represented by a Young diagram (where $\la$ is a nonnegative signature). For example, if $\la=(4,3,1)$, then the corresponding diagram looks as 
\begin{align*}
	\begin{array}{|c|c|c|c|}
	\hline\ &\ &\ &\ \\
	\hline\ &\ &\ &\multicolumn{1}{|c}\ \\
	\cline{1-3}\ \\
	\cline{1-1}
	\end{array}
\end{align*}
Let $d$ be the number of boxes on the diagonal of $\la$ ($d=2$ in the above example), and let $n$ be the total number of boxes in the diagram ($n=8$ in the above example). Show that $d\le \sqrt n$.









\end{document}