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\title{Math 668 Problem Set 1: Due Wednesday, September 7}



\begin{document}

\maketitle
\thispagestyle{empty}
See the course website for homework policy.


\bigskip

\begin{Problem}
Use the mathematical software package of your choice to do these computations:

\begin{enumerate}
\item  Write down the $6$ transition matrices relating the $e$, $h$ and $m$ bases for the degree $4$ symmetric polynomials. (These should be $5 \times 5$ matrices.)

\item Expand $(x_1+x_2)(x_1+x_3)(x_1+x_4)(x_2+x_3)(x_2+x_4)(x_3+x_4)$ in the $e$-basis of $\Lambda_4$. 
\end{enumerate}
\end{Problem}

\begin{Problem}
Let $g \in \GL_n(\CC)$, and let the characteristic polynomial of $g$ factor as $\prod_{i=1}^n (x-\lambda_i)$. 
Let $t$ be the diagonal matrix with entries $\lambda_1$, $\lambda_2$, \dots, $\lambda_n$.
Let $\chi : \GL_n(\CC) \to \CC$ be an continuous function which is constant on conjugacy classes. Show that $\chi(g) = \chi(t)$. 
\end{Problem}

\begin{Problem}
Let $G$ be a group acting on a complex vector space $V$ by a map $\rho : G \to \GL(V)$. Recall that we define the character $\chi_V$ of this action by $\chi_V(g) = \text{Tr} \rho(g)$.
\begin{enumerate}
\item Let $V$ and $W$ be representations of $G$. Define the ``obvious" representation of $G$ on $V \oplus W$ and show that $\chi_{V \oplus W}(g) = \chi_V(g) + \chi_W(g)$.
\item Let $V$ and $W$ be representations of $G$. Define the ``obvious" representation of $G$ on $V \otimes W$ and show that $\chi_{V \otimes W}(g) = \chi_V(g) \times \chi_W(g)$.
\end{enumerate}
For the next several problems, let $G$ be $\GL_3(\CC)$ and let $g = \begin{sbm} x_1&& \\ &x_2& \\ &&x_3 \\ \end{sbm}$.
Let $V$ be the standard representation of $G$ on $\CC^3$.
\begin{enumerate}[resume]
\item Define the ``obvious" representation of $G$ on $\Alt^2 V$ and compute $\chi_{\Alt^2 V}(g)$.
\item Define the ``obvious" representation of $G$ on $\Sym^2 V$ and compute $\chi_{\Sym^2 V}(g)$.
\item Define the ``obvious" representation of $G$ on $V^{\vee}$ and compute $\chi_{V^{\vee}}(g)$.
\end{enumerate}
\end{Problem}

\begin{Problem}
Let $\lambda$ and $\mu$ be two partitions with $|\lambda| = |\mu|$. Show that $\lambda \preceq \mu$ if and only if $\mu^T \preceq \lambda^T$. 
\end{Problem}


%\textbf{Problem 3}
%Let $\phi: GL_2(\CC) \to \CC$ be a continuous funtion which obeys $\phi(g h 
%g^{-1}) = \phi(h)$. Show that $\phi \left( \begin{smallmatrix} \lambda & 1 \\ 
%0 & \lambda \end{smallmatrix} \right) = \phi \left( \begin{smallmatrix} 
%\lambda & 0 \\ 0 & \lambda \end{smallmatrix} \right)$.
%
%\medskip
%
%\textbf{Problem 4} \textbf{\emph{This problem has been modified}} In class, we described the standard inclusion $\Lambda_n 
%\to \Lambda$ by sending $h_{\lambda} \mapsto h_{\lambda}$ for $\ell(\lambda) \leq n$.  Show that this is not a map of rings.
%
%
%\medskip

%\begin{Problem}
%\begin{enumerate}
%\item Show that 
%\[ \prod_{i,j} (1+x_i y_j) = \sum_{\lambda} e_{\lambda}(x) m_{\lambda}(y) . \]
%
%\item Define $b_{\lambda \mu} = \langle e_{\lambda}, h_{\mu} \rangle$. Show that $b_{\lambda \mu} = b_{\mu \lambda}$.
%
%\item Show that the involution $\omega$ preserves $\langle \ , \ \rangle$.
%
%\item In class, we gave an interpretation of $\langle h_{\lambda}, h_{\mu} \rangle$ in terms of matrices of nonnegative integers with given row and column sum. Give a similar interpretation of $b_{\lambda \mu}$.
%\end{enumerate}
%\end{Problem}

\begin{Problem}
We recall from class the notations $e_k$ and $h_k$:
\[ e_k = \sum_{i_1 < i_2 < \cdots < i_k} x_{i_1} x_{i_2} \cdots x_{i_k} \qquad h_k = \sum_{i_1 \leq i_2 \leq \cdots \leq i_k} x_{i_1} x_{i_2} \cdots x_{i_k} . \]
In class we showed that each could be written as a polynomial in the others but didn't give an explicit formula. Remedy this by showing that
\[ h_k = \det \begin{bmatrix} e_1 & e_2 & e_3 & \cdots & e_k \\ 1 & e_1 & e_2 & \cdots & e_{k-1} \\ 0 & 1 & e_1 & \cdots & e_{k-2} \\ &&\ddots&\ddots&\vdots \\ &&& 1 & e_1 \\ \end{bmatrix}  \qquad
 e_k = \det \begin{bmatrix} h_1 & h_2 & h_3 & \cdots & h_k \\ 1 & h_1 & h_2 & \cdots & h_{k-1} \\ 0 & 1 & h_1 & \cdots & h_{k-2} \\ &&\ddots&\ddots&\vdots \\ &&& 1 & h_1 \\ \end{bmatrix}. \]
\end{Problem}

\begin{Problem}
 This problem will work you through the basic properties of 
the power symmetric functions; they are important but won't come up very often for us. 
%It will provide our first (but not best) proof that $\langle \ , \ \rangle$ is positive definite.

Define $p_k(x) = \sum x_i^k$ and define $p_{\lambda}(x) = \prod_i p_{\lambda_i}(x)$.

\begin{enumerate}
\item Prove Newton's Identity:
\[ k e_k = e_{k-1} p_1 - e_{k-2} p_2 + e_{k-3} p_3 - \cdots \pm e_1 p_{k-1} \mp  p_k. \]

\item Show that $\Lambda \otimes \QQ$ is $\QQ[p_1, p_2, p_3, \ldots]$.
%
%\item  Establish a formula of the form
%\[  \prod_{i,j} \frac{1}{1-x_i y_j} = \sum_{\lambda} z_{\lambda} p_{\lambda}(x) p_{\lambda}(y) . \]
%for some positive rational numbers $z_{\lambda}$. Hint: $\frac{1}{1-w} = \exp (w+w^2/2+w^3/3+w^4/4+\cdots)$.
%
%\item Compute $\langle p_{\lambda}, p_{\mu} \rangle$ and deduce that $\langle \ , \ \rangle$ is positive definite.
\end{enumerate}
\end{Problem}

\end{document}
