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\begin{document}

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\centerline{\sc Problem Set Five: Due Friday, October 23}


\begin{Problem}
Let $M$ be a matrix. We define a top justified consecutive minor of $M$ to be a minor of the form $\Delta^{12\cdots}_{(a+1)(a+2) \cdots (a+k)}(M)$, for some $a$ and $k$, and a left justified consecutive minor to be a minor of the form $\Delta^{(b+1)(b+2) \cdots (b+k)}_{12 \cdots k}(M)$ for some $b$ and $k$. We'll define a top-or-left justified minor to be a minor which is either top or left justified (or both).

\begin{enumerate}
\item Show that a totally positive matrix is determined by the values of its top-or-left justified minors.
\end{enumerate}

Let $X$ be a $n \times n$ totally positive matrix. Let $D$ be the diagonal matrix with entries 
\[ \left( \Delta^1_1(X), \frac{\Delta^{12}_{12}(X)}{\Delta^1_1(X)}, \frac{\Delta^{123}_{123}(X)}{\Delta^{12}_{12}(X)}, \ldots, \frac{\Delta^{[n]}{[n]}(X)}{\Delta^{[n-1]}_{[n-1]}(X)} \right) . \]
Let $L$ be the unique lower triangular matrix such that
\[ \Delta^{(b+1)(b+2) \cdots (b+k)}_{12 \cdots k}(L) =  \frac{\Delta^{(b+1)(b+2) \cdots (b+k)}_{12 \cdots k}(X)}{\Delta^{12 \cdots k}_{12 \cdots k}(X)} \]
and let $U$ be the unique upper triangular matrix such that
\[ \Delta^{12 \cdots k}_{(a+1)(a+2) \cdots (a+k)}(U) =  \frac{\Delta^{12 \cdots k}_{(a+1)(a+2) \cdots (a+k)}(X)}{\Delta^{12 \cdots k}_{12 \cdots k}(X)}. \]

\begin{enumerate}[resume]
\item Show that $X = LDU$.
\end{enumerate}
We have now confirmed the claim from earlier in the course that, if $X$ is totally positive and has $LDU$ factorization $(L, D, U)$, then $L$, $D$ and $U$ are totally nonnegative.
\end{Problem}

\begin{Problem}
The Pl\"ucker coordinates on the Grassmannian $G(2,4)$ obey the relation $\Delta^{13} \Delta^{24} = \Delta^{12} \Delta^{34} + \Delta^{14} \Delta^{23}$. A point of $G(2,4)$ is called totally nonnegative if all of its Pl\"ucker coordinates are nonnegative.
\begin{enumerate}
\item Which of the subsets of $\{ 12, 13, 14, 23, 24, 34 \}$ are capable of being the sets $\{ I : \Delta^I(x) \neq 0 \}$ for $x$ a totally nonnegative point of $G(2,4)$? You should find that there are $33$ in total.
\item Give examples of points of $G(2,4)$ where each of the following occurs
\[ \Delta^{12}, \Delta^{13}, \Delta^{14}, \Delta^{23}, \Delta^{24}, \Delta^{34}  > 0 \]
\[ \Delta^{12}, \Delta^{13}, \Delta^{14}, \Delta^{23}, \Delta^{24} > 0,\quad \Delta^{34} = 0 \]
\[\Delta^{13}, \Delta^{14}, \Delta^{23}, \Delta^{24} > 0,\quad \Delta^{12} =  \Delta^{34} = 0 \]
\[ \Delta^{12}, \Delta^{13}, \Delta^{14} > 0 \quad \Delta^{23} = \Delta^{24}= \Delta^{34} = 0 \]
\[ \Delta^{12}, \Delta^{13}, \Delta^{23} > 0 \quad \Delta^{14} = \Delta^{24}= \Delta^{34} = 0 \]
\end{enumerate}


\end{Problem}

%LU factorization
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%G(2,4) positroids

\end{document}


