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\title{Math 668 Problem Set 9: \\ Due Friday, November 18 \\ Last problem set!}





\begin{document}

\maketitle
\thispagestyle{empty}

\begin{Problem}
Let $V_{311}$ be the irreducible representation of $\GL_3$ with character $s_{31}(x_1, x_2, x_3)$. We'll write $\rho : \GL_3 \to \GL(V_{31})$ for the representation homomorphism.
\begin{enumerate}
\item Write down a basis of $V_{31}$ in terms of your favorite construction of $V_{31}$. Hint: $\dim V_{31}$ is $15$.
\item Write down the action of $\rho \left( \begin{sbm} 1&u&0 \\ 0&1&0 \\ 0&0&1 \\ \end{sbm} \right)$ on your basis. (It should break into many small blocks, so there isn't as much to write as you might fear.
\item Let $r : \Mat_{3 \times 3} \to \End(V)$ be the corresponding Lie algebra map. Write down the  action of $\rho \left( \begin{sbm} 0&1&0 \\ 0&0&0 \\ 0&0&0 \\ \end{sbm} \right)$ on your basis. 
\end{enumerate}
\end{Problem}

\begin{Problem}
Let $\lambda$ be a partition of $n$. We first review our construction of $V_{\lambda}$ in terms of products of matrix minors.

We work with $n \times n$ matrices whose entries are variables $z_{ij}$. 
For $J = \{ j_1, j_2, \ldots, j_k \}$ a $k$-element subset of $[n]$, let
\[ \Delta_J = \det \begin{bmatrix}
z_{1 j_1} & z_{1 j_2} & \cdots & z_{1 j_k} \\
z_{2 j_1} & z_{2 j_2} & \cdots & z_{2 j_k} \\
\vdots & \vdots & \ddots & \vdots \\
z_{k j_1} & z_{k j_2} & \cdots & z_{k j_k} \\
\end{bmatrix} . \]
If $T$ is a tableau (semistandard or not) with columns $J^1$, $J^2$, \dots, $J^m$, then we put $\Delta(T) = \prod_j \Delta_{J^j}$. 
We showed that the span of $\Delta(T)$, with $T$ of shape $\lambda$, is $V_{\lambda}$. In this problem, we will show that the $\Delta(T)$ with $T$ semi-standard form a basis of $V_{\lambda}$.
\begin{enumerate}
\item To check that you understand the definitions, write down 
\[ \Delta\left(\young(12,3)\right),\  \Delta\left(\young(13,2)\right),\  \Delta\left(\young(21,3)\right),\  \Delta\left(\young(12,34)\right),\  \Delta\left(\young(13,24)\right) \ \text{and}\   \Delta\left(\young(12,43)\right)  \]
explicitly as polynomials in the $z_{ij}$.
\item Write $ \Delta\left(\young(21,3)\right)$ as a linear combination of $\Delta\left(\young(12,3)\right)$ and $\Delta\left(\young(13,2)\right)$. 
Write $\Delta\left(\young(12,43)\right)$ as a linear combination of $ \Delta\left(\young(12,34)\right)$ and $\Delta\left(\young(13,24)\right)$.
\end{enumerate}
Choose any matrix $w_{ij}$  of positive integers such that, for $i_1 < i_2$ and $j_1 < j_2$, we always have $w_{i_1 j_1} + w_{i_2 j_2} > w_{i_1 j_2} + w_{i_2 j_1}$. An explicit example is to take $w_{ij} = ij$. 
Put an order on the set of monomials $\{ \prod z_{ij}^{A_{ij}} \}$ by defining $ \prod z_{ij}^{A_{ij}} \succ  \prod z_{ij}^{B_{ij}}$ if $\sum A_{ij} w_{ij} > \sum B_{ij} w_{ij}$ (and breaking ties arbitrarilary). 
For a nonzero polynomial $f \in \CC[z_{11}, z_{12} , \ldots, z_{nn} ]$, we define the \newword{leading monomial} of $f$ to be the largest monomial with nonzero coefficient in $f$.
\begin{enumerate}[resume]
\item Let $T$ be a tableau with strictly increasing columns. Describe the leading term of $\Delta(T)$.
\item Show that, if $T$ and $U$ are distinct semistandard Young tableaux, then $\Delta(T)$ and $\Delta(U)$ have different terms. Conclude that $\{ \Delta(T) : T \in \SSYT(\lambda) \}$ is a basis for $V_{\lambda}$. 
\end{enumerate}
\end{Problem}

\end{document}