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\begin{document}

\thispagestyle{empty}
\centerline{\sc Problem Set Six: Due Friday, October 30}

\begin{Problem}
Cancelled, because the main claim was in error.
\end{Problem}

\begin{Problem}
Let
\[ \dot{s}_i = \begin{sbm} 
1&&          && &&\\
&1&          && &&\\
&&\ddots  && &&\\
&&             &0&1 &&\\
&&             &-1&0 &&\\
&&             &&&\ddots& \\
&&             &&&&1 \\
\end{sbm} \]
where the $2 \times 2$ block is in rows and columns $i$ and $i+1$.
\begin{enumerate}
\item For $w \in S_n$, let $s_{i_1} s_{i_2} \cdots s_{i_N}$ be a reduced word for $w$. Define $\dot{w}$ to be $\dot{s}_{i_1} \dot{s}_{i_2} \cdots \dot{s}_{i_N}$. Show that $\dot{w}$ is independent of the choice of reduced word.
\item Show that each nonvanishing right justified minor of $\dot{w}$ is $1$. 
\end{enumerate}
In class, we showed that, for $g \in N_+ \cap B_- w B_-$, there is a unique $f \in N_- w \cap w N_+$ with $g B_- = f B_-$. The same argument shows that there is a unique $\dot{f} \in N_- \dot{w} \cap \dot{w} N_+$ with $g B_- = \dot{f} B_-$. (You don't have to check this.) This $\dot{f}$ is slightly more useful than $f$, for the following reason:
\begin{enumerate}[resume]
\item Suppose that $g$ is totally nonnegative. Show that each right justified minor of $\dot{f}$ is nonnegative.
\end{enumerate}
\end{Problem}

\begin{Problem}
What paper/area would you like to learn about for your final presentation?
\end{Problem}


%
%We remind you that, if a wiring diagram has a strand that starts at height $i$ and then goes to position $j$, then the corresponding permutation matrix $w$ has a $1$ in position $(i,j)$. 
%
%\begin{Problem} \label{MakeConj}
%Make a more precise conjecture: The matrix entry in $f$ which is a function of $t_1$, $t_2$, \dots, $t_r$ (only) occurs in which position of the matrix?
%\end{Problem}
%
%This is a case where I found it more natural to work from the left than the right. Let $w'_r = s_{i_2} s_{i_3} \cdots s_{i_r}$, let $g'_r = x_{i_2}(t_2) \cdots x_{i_r}(t_r)$ and let $f'_r$ be the unique matrix in $N_- w'_r \cap w'_r N_+$ with $g'_r B_- = f'_r B_-$. Abbreviate $i_1 = i$ and $t_1 = t$. 
%Let the strands of the $w'_r$ wiring diagram that start at $i$ and $i+1$ end at $j_1$ and $j_2$.
%
%\begin{Problem}
%What do the matrices $g_r$, $g'_r$, $f_r$ and $f'_r$ look like in rows $i$ and $i+1$, and columns $j_1$ and $j_2$?
%Show, in particular, that $(f_r)_{(i+1) j_2} = t_1^{-1}$.
%\end{Problem}
%
%\begin{Problem}
%Prove the rest of your conjecture from Problem~\ref{MakeConj}.
%\end{Problem}
%



\end{document}


