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\title{Math 668 Problem Set 8: \\ Due Friday, November 11}





\begin{document}

\maketitle
\thispagestyle{empty}

\begin{Problem}
\begin{enumerate}
\item Write down bases for $\Alt^2 \CC^3$ and $\Sym^2 \CC^2$.
\item Write down the matrix for the natural action of $g = \begin{sbm} g_{11} & g_{12} & g_{13} \\ g_{21} & g_{22} & g_{23} \\ g_{31} & g_{32} & g_{33} \\ \end{sbm}$ on  $\Alt^2 \CC^3$.
\item Write down the matrix for the natural action of $g = \begin{sbm} g_{11} & g_{12} \\ g_{21} & g_{22}  \\  \end{sbm}$ on  $\Sym^2 \CC^2$.
\end{enumerate}
\end{Problem}


\begin{Problem}
This problem works through the main results about Schur-Weyl duality, which concerns the relationship between $\GL_n(\CC)$ and $S_n$ representations. 
Let $T = \begin{sbm} t_1 && \\ &\ddots& \\ && t_n \\ \end{sbm}$ be the torus in $\GL_n$.
If $V$ is any algebraic representation of $\GL_n$, then let $V_{1 \cdots 1}$ be the subspace on which $T$ acts by the character $(1,1, \ldots, 1)$. (This may be $0$-dimensional.)
\begin{enumerate}
\item Show that $S_n \subset \GL_n$ maps $V_1$ to itself.
\item Let $\lambda$ be a partition and let $V_{\lambda}$ be the corresponding polynomial representation of $V_{\lambda}$. Show that $(V_{\lambda})_{1 \cdots 1}$ is nonzero if and only if $|\lambda| = n$.
For $\lambda$ a partition of $n$, define $W_{\lambda} = (V_{\lambda})_{1 \cdots 1}$.
\end{enumerate}
Let $R$ be the $n^2$-dimensional polynomial ring $\CC[x_{11}, x_{12}, \ldots, x_{nn}]$. I'll write ${}_{1 \cdots 1} R_{1 \cdots 1}$ for the subspace of $R$ where $T$ acts by character $(1,1,\ldots,1)$ for both the left and right action of $T$, and I'll write $R_{1 \cdots 1}$ for the subspace where $T$ acts by $(1,\ldots, 1)$ from the right. 
\begin{enumerate}[resume]
\item Show that ${}_{1 \cdots 1} R_{1 \cdots 1} \cong \CC S_n$ as an $S_n \times S_n$ representation. Show that $R_{1 \cdots 1} \cong (\CC^n)^{\otimes n}$ as a $\GL_n \times S_n$ represenation.
\item Show that ${}_{1 \cdots 1} R_{1 \cdots 1}  \cong \bigoplus_{|\lambda|=n} W_{\lambda} \otimes W_{\lambda}^{\vee}$ as an $S_n \times S_n$ representation. Show that $R_{1 \cdots 1} \cong \bigoplus_{|\lambda|=n} V_{\lambda} \otimes W_{\lambda}^{\vee}$ as a $\GL_n \times S_n$ representation.
\item Using the identity $\CC S_n \cong {}_{1 \cdots 1} R_{1 \cdots 1}  \cong \bigoplus_{|\lambda|=n} W_{\lambda} \otimes W_{\lambda}^{\vee}$, show that the $W_{\lambda}$ are a complete list of the irreducible representations of $S_n$.
\end{enumerate}
%For each partition $\lambda$ of $n$, let $V_{\lambda}$ be the irreducible representation of $\GL_n(\CC)$ with character $s_{\lambda}(x_1, \ldots, x_n)$. 
%Let $W_{\lambda}$ be the subspace on $V_{\lambda}$ on which the torus $T$ acts by the character
\end{Problem}

\begin{Problem}
This problem studies the Young symmetrizer. Let $\lambda$ be a partition of $n$. Choose a numbering of the boxes of $\lambda$ by $[n]$. Let $S_R \subset S_n$ be the subgroup which permutes the boxes within rows, so $S_R \cong \prod S_{\lambda_k}$; let $S_C$ be  the subgroup which permutes the boxes within columns, so $S_C \cong \prod S_{\lambda^T_k}$. We begin with some pure combinatorics:
\begin{enumerate}
\item Let $\pi$ be a permutation in $S_n$. Show that exactly one of the following is true: Either $\pi$ factors as $\rho \sigma$ for $\rho \in S_R$ and $\sigma \in S_C$, or else there are two boxes $i$ and $j$ in the same column such that $\pi(i)$ and $\pi(j)$ are in the same row.
\end{enumerate}
We define $\alpha_{\lambda} = \sum_{\sigma \in S_C} (-1)^{\sigma} \sigma$ and $\beta_{\lambda} = \sum_{\rho \in S_R} \rho$, as elements of the group algebra.
\begin{enumerate}[resume]
\item Show that, for any permutation $\pi \in S_n$, we either have $\beta_{\lambda} \pi \alpha_{\lambda} = \pm \beta_{\lambda} \alpha_{\lambda}$, or else $\beta_{\lambda} \pi \alpha_{\lambda} = 0$. 
\item The Young symmetrizer is defined as $\omega_{\lambda} := \beta_{\lambda} \alpha_{\lambda}$. Show that $\omega_{\lambda}^2$ is a scalar multiple of $\omega_{\lambda}$. 
\end{enumerate}
\end{Problem}

\end{document}