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\usepackage{amssymb, amsmath, amsfonts, amsthm, graphics, epstopdf}
\usepackage[hmargin=1 in, vmargin = 0.5 in]{geometry}
\usepackage{youngtab}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}
\usepackage[normalem]{ulem}


% When you first define a new word, use this macro to make it stand out
% EG We say that an abelian group $I$ is \newword{injective} if, for any
% injection $G \to H$, and any map $G \to I$, there is a map $H \to I$ making the
% obvious diagram commute.
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%Arrows
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%Things LaTeX names by appearance, rather than meaning
% By now, I've learned the standard LaTeX names, but I remember they used to give me trouble, so here are some macros
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%Their calligraphic versions; use these for the sheaf constructions
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%This makes alternating tensors look right in displayed equations
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\theoremstyle{definition}
\newtheorem{Problem}{Problem}


\pagestyle{empty}


\title{Math 668 Problem Set 4: Due Wednesday, September 28}


%% NOTE TO SELF: Almost all the problems from 2012 PSet 3 aren't appropriate yet but will be in a week or two; check in then.



\begin{document}

\maketitle
\thispagestyle{empty}

\begin{Problem}
Let $\lambda$ be a partition and put $\delta(\lambda, d) = s_{\lambda}(1,1,1, \ldots, 1)$ where there are $d$ ones. 
\begin{enumerate}
\item Compute $\delta(2,d)$, $\delta((1,1), d)$, $\delta((2,1), d)$ and $\delta((2,2), d)$.
\item Show that $\delta(\lambda, d)$ is a polynomial in $d$.
\end{enumerate}
\end{Problem}


\begin{Problem}
The goal of this problem is to get you to fill in the details of the proof of the dual Jacobi-Trudi identity, which asserts that
\[ s_{\lambda^T} = \det \begin{bmatrix}
e_{\lambda_1} & e_{\lambda_1+1} & e_{\lambda_1+2} & \cdots & e_{\lambda_1+\ell} \\
e_{\lambda_2-1} & e_{\lambda_2} & e_{\lambda_2+1} & \cdots & e_{\lambda_2+\ell-1} \\
e_{\lambda_3-2} & e_{\lambda_3-1} & e_{\lambda_3} & \cdots & e_{\lambda_3+\ell-2} \\
\vdots&\vdots&\vdots&\ddots&\vdots \\
e_{\lambda_{\ell} - \ell+1}&e_{\lambda_{\ell} - \ell+2}&e_{\lambda_{\ell} - \ell+3}&\cdots&e_{\lambda_{\ell}} \\
\end{bmatrix}. \]
\begin{enumerate}
\item Take a $k \times (n-k)$ rectangular grid and direct all the edges up and right. The figure below depicts $k=2$, $n=5$.

\centerline{
\begin{tikzpicture}
\draw[-{Latex[length=3mm, width=2mm]}] (0,0) -- (1,0) ;\draw[-{Latex[length=3mm, width=2mm]}] (1,0) -- (2,0) ; \draw[-{Latex[length=3mm, width=2mm]}] (2,0) -- (3,0) ;
\draw[-{Latex[length=3mm, width=2mm]}] (0,1) -- (1,1) ;\draw[-{Latex[length=3mm, width=2mm]}] (1,1) -- (2,1) ; \draw[-{Latex[length=3mm, width=2mm]}] (2,1) -- (3,1) ;
\draw[-{Latex[length=3mm, width=2mm]}] (0,2) -- (1,2) ;\draw[-{Latex[length=3mm, width=2mm]}] (1,2) -- (2,2) ; \draw[-{Latex[length=3mm, width=2mm]}] (2,2) -- (3,2) ;
\draw[-{Latex[length=3mm, width=2mm]}] (0,0) -- (0,1) ; \draw[-{Latex[length=3mm, width=2mm]}] (0,1) -- (0,2) ; 
\draw[-{Latex[length=3mm, width=2mm]}] (1,0) -- (1,1) ; \draw[-{Latex[length=3mm, width=2mm]}] (1,1) -- (1,2) ; 
\draw[-{Latex[length=3mm, width=2mm]}] (2,0) -- (2,1) ; \draw[-{Latex[length=3mm, width=2mm]}] (2,1) -- (2,2) ; 
\draw[-{Latex[length=3mm, width=2mm]}] (3,0) -- (3,1) ; \draw[-{Latex[length=3mm, width=2mm]}] (3,1) -- (3,2) ; 
\end{tikzpicture}} \noindent
Show how to put weights on the edges of this graph such that the weighted sum of all paths from the lower left to the upper right is $e_k(x_1, x_2, \ldots, x_n)$.

\item Prove the dual Jacobi-Trudi identity.
\end{enumerate}
\end{Problem}

\begin{Problem}
The point of this problem is to give a direct proof of the dual Cauchy identity:
\[ \prod_{i,j=1}^n (1+x_i y_j) = \sum_{\lambda} s_{\lambda}(x_1, x_2,\ldots, x_n) s_{\lambda^T}(y_1, y_2, \ldots, y_n). \]
%(You could also deduce this by putting together other things you have done, but let's prove it directly.)
%Recall the definitions of $\uparrow$ and $\downarrow$ from class. 
Define $\alpha \rightarrow \beta$ to mean $\alpha^T \uparrow \beta^T$, and $\alpha \leftarrow \beta$ to mean $\alpha^T \downarrow \beta^T$.
\begin{enumerate}
\item Show that, for two partitions $\alpha$ and $\beta$, we have $\alpha \rightarrow \beta$ iff $\alpha_j \leq \beta \leq \alpha_j+1$ for all $j$.
\item Fix partitions $\beta$ and $\gamma$. Show that
\[ \sum_{\beta \uparrow \delta \leftarrow \gamma} x^{|\delta| - |\beta|} y^{|\delta| - |\gamma|} = (1+xy) \sum_{\beta \leftarrow \alpha \uparrow \gamma} y^{|\beta| - |\alpha|} x^{|\gamma| - |\alpha|} . \]
\item Prove the dual Cauchy identity.
\end{enumerate}
\end{Problem}

\begin{Problem}
Problem 4 is removed because of flawed wording in a boundary case. It will return next week.
%Here is a different, more visual way to describe the Jacobi-Trudi identity.
%Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_{\ell})$ be a partition. We define a \newword{special rim hook tableau} to be a labeling of the boxes of $\lambda$ with the colors $1$, $2$, \ldots, $\ell$ such that 
%\begin{enumerate}
%\item The labels in the first column are $\{ 1,2,3, \ldots, \ell \}$ in order and 
%\item The boxes of each given number form a connected shape traveling up and to the left, with no $2 \times 2$ subblocks.
%\end{enumerate}
%For example, here are the $6$ special rim hook tableau of shape $(6,5,3)$:
%\[ \young(111111,22222,333),\ \young(111122,22222,333), \young(111111,22333,333), \young(122222,22333,333),  \young(111133,22333,333),  \young(122233,22333,333) . \]
%Let $c_j(T)$ be the number of $j$'s in $T$, and let $h(T)$ be the number of places where $T$ has two vertically stacked boxes with the same entry.
%Show that the Jacobi-Trudi identity is equivalent to the formula:
%\[ s_{\lambda} = \sum_T (-1)^{h(T)} h_{c_1(T)} h_{c_2(T)} \cdots h_{c_{\ell}(T)} . \]

\end{Problem}



\end{document}