\documentclass[11pt]{amsart}
\usepackage{amssymb, amsmath, amsfonts, amsthm, graphics, epstopdf}
\usepackage[hmargin=1 in, vmargin = 0.5 in]{geometry}
\usepackage{youngtab}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}


% 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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\newcommand{\bigintersection}{\bigcap}
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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
\newcommand{\Alt}{\bigwedge\nolimits}


%Blackboard bold letters.
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\newcommand{\SSYT}{\text{SSYT}}


\theoremstyle{definition}
\newtheorem{Problem}{Problem}


\pagestyle{empty}


\title{Math 668 Problem Set 3: Due Wednesday, September 21}


%% 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}

See the course website for homework policy.



\bigskip

\begin{Problem}
Give the six $5 \times 5$ matrices which convert between the Schur polynomials $s_{\lambda}$ of degree $4$, and our other favorite bases $m_{\lambda}$, $e_{\lambda}$, $h_{\lambda}$. (Using technology is encouraged.)
\end{Problem}

Let $\lambda$ be a partition and let $n$ be a positive integer. Let $\SSYT(\lambda, n)$ be the set of semistandard young Tableaux of shape $\lambda$ with entries from $\{ 1,2,\ldots, n \}$. 
In the next few problems, we will give other objects which are in bijection with $\SSYT(\lambda, n)$.

\begin{Problem}
List the eight elements of $\SSYT((2,1), 3)$.
\end{Problem}

\begin{Problem}
A Gelfand-Tsetlin pattern is an array of $\binom{n+1}{2}$ nonnegative integers $g_ij$, for $1 \leq j \leq i \leq n$, obeying the inequalities
\[ g_{ij} \geq g_{(i-1)j} \geq g_{i(j+1)} .\]
We will display them in a triangular array:
\[ \begin{bmatrix}
g_{11} &&&& \\
g_{21} & g_{22} &&& \\
g_{31} & g_{32} & g_{33} && \\
\vdots & & \ddots & \\
g_{n1} & g_{n2} & g_{n3} & \cdots & g_{nn} \\ 
\end{bmatrix}. \]

\begin{enumerate}
\item List the eight Gelfand-Tsetlin patterns with $(g_{31}, g_{32}, g_{33}) = (2,1,0)$.
\item Give a bijection between $\SSYT(\lambda, n)$ and $n \times n$ Gelfand-Tsetlin patterns with $g_{nj} = \lambda_j$.
\end{enumerate}
\end{Problem}

\begin{Problem}
Define a directed graph whose vertices are the integer lattice points $(i,j)$, and where we have edges $(i,j) \longrightarrow (i+1, j)$ and $(i,j) \longrightarrow (i, j+1)$. 
A \newword{directed path} is a path through this graph which follows the directions of the arrows.
A \newword{collection of vertex disjoint directed paths} is a collection of directed paths so that no two have any vertices in common.
\begin{enumerate}
\item List the eight pairs of vertex disjoint paths $\gamma_1$, $\gamma_2$, where $\gamma_1$ goes from $(1,-1)$ to $(3,1)$ and $\gamma_2$ goes from $(1,-2)$ to $(3,-1)$. In the diagram below, the starting locations are shown as white circles and the ending locations as black circles.

\centerline{\begin{tikzpicture}
\draw[-{Latex[length=3mm,width=3mm]}, thick] (1,-2) -- (2,-2) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (2,-2) -- (3,-2) ;
\draw[-{Latex[length=3mm,width=3mm]}, thick] (1,-1) -- (2,-1) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (2,-1) -- (3,-1) ;
\draw[-{Latex[length=3mm,width=3mm]}, thick] (1,0) -- (2,0) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (2,0) -- (3,0) ;
\draw[-{Latex[length=3mm,width=3mm]}, thick] (1,1) -- (2,1) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (2,1) -- (3,1) ;
\draw[-{Latex[length=3mm,width=3mm]}, thick] (1,-2) -- (1,-1) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (1,-1) -- (1,0) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (1,0) -- (1,1) ;
\draw[-{Latex[length=3mm,width=3mm]}, thick] (2,-2) -- (2,-1) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (2,-1) -- (2,0) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (2,0) -- (2,1) ;
\draw[-{Latex[length=3mm,width=3mm]}, thick] (3,-2) -- (3,-1) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (3,-1) -- (3,0) ; \draw[-{Latex[length=3mm,width=3mm]}, thick] (3,0) -- (3,1) ;
\filldraw[color=black, fill=white] (1,-1) circle (0.15); \filldraw[color=black, fill=white] (1,-2) circle (0.15);
\filldraw[color=black, fill=black] (3,1) circle (0.15); \filldraw[color=black, fill=black] (3,-1) circle (0.15);
\end{tikzpicture}}
\item Give a bijection between $\SSYT$ and $n$-tuples of vertex disjoint paths $(\gamma_1, \gamma_2, \ldots, \gamma_n)$ where $\gamma_i$ connects $(1, -i)$ to $(n, \lambda_i - i)$. 
\end{enumerate}
\end{Problem}

\begin{Problem}
Let $\epsilon : S_n \to \{ \pm 1 \}$ be the map which sends a permutation to its sign. A polynomial $f(x_1, x_2, \ldots, x_n)$ in $\ZZ[x_1, x_2, \ldots, x_n]$ is called \newword{alternating} if 
\[ f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)}) =\epsilon(\sigma) f(x_1, x_2, \ldots, x_n) \]
for all $\sigma \in S_n$. Put $\Delta := \prod_{1 \leq i < j \leq n} (x_i - x_j)$. 

Show that a polynomial $f(x_1, x_2, \ldots, x_n)$ in $\ZZ[x_1, x_2, \ldots, x_n]$ is alternating if and only if $f(x_1, x_2, \ldots, x_n)$ is of the form $\Delta(x_1, x_2, \ldots, x_n) g(x_1, x_2, \ldots, x_n)$ for some symmetric polynomial $g$.
\end{Problem}



\end{document}