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\title{Math 668 Problem Set 2: Due Wednesday, September 14}



\begin{document}

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


\bigskip

\begin{Problem} \label{Omega}
We have shown that $\Lambda  = \ZZ[h_1, h_2, h_3, \ldots] = \ZZ[e_1, e_2, \ldots]$. 
Define  $\omega : \Lambda \to \Lambda$ to be the unique ring homomorphism such that $\omega(h_n) = e_n$.
Show that $\omega(e_n) = h_n$ and thus $\omega^2$ is the identity map.
\end{Problem} 

\begin{Problem}
Show that $\bigoplus_{\ell(\lambda) \leq n} \ZZ h_{\lambda}$ is \textbf{not} a subring of $\Lambda$.
\end{Problem}


\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 $\omega$, from Problem~1, 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}
This problem follows up on our previous problem on power sum symmetric functions, and you may use results from that problem.
\begin{enumerate}
\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}

\begin{Problem}
Let $1^k$ be the partition $11 \cdots 1$ with $k$ ones.
\begin{enumerate}
\item Show that $\langle h_{1^k}, h_{1^k} \rangle = k!$.
\end{enumerate}
Recall that, for $G$ a group and $V$ and $W$ two representations of $G$, the space $\Hom^G(V,W)$ is the space of linear maps $f : V \to W$ such that $f(g \cdot \vec{v})  = g \cdot f(\vec{v})$ for all $g \in G$ and $\vec{v} \in V$. I promised you $\langle \ , \ \rangle$ would compute the dimension of the corresponding $\Hom$ space. In this case, my promise is that we should have $\dim \Hom^{\GL_n}((\CC^n)^{\otimes k}, (\CC^n)^{\otimes k}) = k!$, at least for $k$ large.
\begin{enumerate}[resume]
\item Give $k!$ elements of $\Hom^{\GL_n}((\CC^n)^{\otimes k}, (\CC^n)^{\otimes k})$ and show that these elements are linearly independent if $n>k$. (Hint: If you choose the elements I expect you to, consider what they do to $e_1 \otimes e_2 \otimes \cdots \otimes e_k$, where $e_1$, $e_2$, \dots, $e_n$ is a basis of $\CC^n$.)
\item If $n<k$, give a linear dependence between your $k!$ elements of  $\Hom^{\GL_n}((\CC^n)^{\otimes k}, (\CC^n)^{\otimes k})$.
\end{enumerate}
\end{Problem}

\end{document}
