Advanced handouts

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Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>

src/Advanced/Symmetric Groups/parts/0 intro.tex

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+\section{Introduction}
+
+
+\definition{}
+Informally, a \textit{permutation} of a collection of $n$ objects is an ordering of these $n$ objects. \par
+For example, a few permutations of $\texttt{A}, \texttt{B}, \texttt{C}, \texttt{D}$ are $\texttt{ABCD}$,
+$\texttt{BCDA}$, and $\texttt{DACB}$. \par
+
+\vspace{2mm}
+
+This, however, isn't the definition we'll use today. Instead of defining permutations as \say{ordered lists,}
+(as we do above), we'll define them as functions. Our first goal today is to make sense of this definition.
+
+
+
+\definition{Permutations}
+Let $\Omega$ be an arbitrary set of $n$ objects. \par
+A \textit{permutation} on $\Omega$ is a map from $\Omega$ to itself that produces a \textit{unique} output for each input. \par
+\note{In other words, if $a$ and $b$ are different, $f(a)$ and $f(b)$ must also be different.}
+
+
+\footnotetext{The words \say{function} and \say{map} are equivalent.}
+
+\vspace{2mm}
+
+For example, consider $\{1, 2, 3\}$. \par
+One permutation on this set can be defined as follows: \par
+\begin{itemize}
+	\item $f(1) = 3$
+	\item $f(2) = 1$
+	\item $f(3) = 2$
+\end{itemize}
+
+If we take the array $123$ and apply
+
+
+\problem{}
+List all permutations on three objects. \par
+How many permutations of $n$ objects are there?
+
+\vfill
+
+
+\problem{}
+What map corresponds to the permutation $[321]$?
+
+
+\vfill
+
+\problem{}
+What map corresponds to the \say{do-nothing} permutation? \par
+Write it as a function and in square-bracket notation. \par
+\note[Note]{We usually call this the \textit{trivial permutation}}
+
+
+\vfill
+\pagebreak
+
+
+We can visualize permutations with a \textit{string diagram}, shown below. \par
+The arrows in this diagram denote the image of $f$ for each possible input.
+Two examples are below:
+
+\vspace{2mm}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {1};
+	\node (2a) at (1, 0.5) {2};
+	\node (3a) at (2, 0.5) {3};
+	\node (4a) at (3, 0.5) {4};
+
+	\node (1b) at (0, -2) {1};
+	\node (3b) at (1, -2) {3};
+	\node (4b) at (2, -2) {4};
+	\node (2b) at (3, -2) {2};
+
+	\line{1a}{1b}
+	\line{2a}{2b}
+	\line{3a}{3b}
+	\line{4a}{4b}
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {1};
+	\node (2a) at (1, 0.5) {2};
+	\node (3a) at (2, 0.5) {3};
+	\node (4a) at (3, 0.5) {4};
+
+	\node (2b) at (0, -2) {2};
+	\node (1b) at (1, -2) {1};
+	\node (3b) at (2, -2) {3};
+	\node (4b) at (3, -2) {4};
+
+	\line{1a}{1b}
+	\line{2a}{2b}
+	\line{3a}{3b}
+	\line{4a}{4b}
+\end{tikzpicture}
+\hfill\null
+\vspace{2mm}
+
+
+Note that in all our examples thus far, the objects in our set have an implicit order.
+This is only for convenience. The elements of $\Omega$ are not ordered (it is a \textit{set}, after all),
+and we may present them however we wish.
+
+
+
+
+
+
+
+\vspace{1cm}
+
+For example, consider the diagrams below. \par
+On the left, 1234 are ordered as usual. In the middle, they are ordered alphabetically. \par
+The rightmost diagram uses arbitrary, meaningless labels.
+
+\vspace{2mm}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {1};
+	\node (2a) at (1, 0.5) {2};
+	\node (3a) at (2, 0.5) {3};
+	\node (4a) at (3, 0.5) {4};
+
+	\node (2b) at (0, -2) {2};
+	\node (1b) at (1, -2) {1};
+	\node (3b) at (2, -2) {3};
+	\node (4b) at (3, -2) {4};
+
+	\line{1a}{1b}
+	\line{2a}{2b}
+	\line{3a}{3b}
+	\line{4a}{4b}
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (4a) at (0, 0.5) {4};
+	\node (1a) at (1, 0.5) {1};
+	\node (3a) at (2, 0.5) {3};
+	\node (2a) at (3, 0.5) {2};
+
+	\node (1b) at (0, -2) {1};
+	\node (4b) at (1, -2) {4};
+	\node (3b) at (2, -2) {3};
+	\node (2b) at (3, -2) {2};
+
+	\line{1a}{1b}
+	\line{2a}{2b}
+	\line{3a}{3b}
+	\line{4a}{4b}
+\end{tikzpicture}
+\hfill
+\begin{tikzpicture}[scale=0.5]
+	\node (1a) at (0, 0.5) {$\triangle$};
+	\node (2a) at (1, 0.5) {$\divideontimes$};
+	\node (3a) at (2, 0.5) {$\circledcirc$};
+	\node (4a) at (3, 0.5) {$\boxdot$};
+
+	\node (2b) at (0, -2) {$\divideontimes$};
+	\node (1b) at (1, -2) {$\triangle$};
+	\node (3b) at (2, -2) {$\circledcirc$};
+	\node (4b) at (3, -2) {$\boxdot$};
+
+	\line{1a}{1b}
+	\line{2a}{2b}
+	\line{3a}{3b}
+	\line{4a}{4b}
+\end{tikzpicture}
+\hfill\null
+\vspace{2mm}
+
+
+It shouldn't be hard to see that despite the different \say{output} order (2134 and 1432), \par
+the same permutation is depicted in all three diagrams. This example demonstrates two things:
+\begin{itemize}[itemsep=2mm]
+	\item First, the names of the items in our set do not have any meaning. \par
+	$\Omega$ is just a set of $n$ arbitrary things, which we may label however we like.
+
+	\item Second, permutations are verbs. We do not care about the \say{output} of a certain permutation,
+	we care about what it \textit{does}. We could, for example, describe the permutation above as
+	\say{swap the first two of four elements.}
+\end{itemize}
+
+\vspace{2mm}
+
+
+Why, then, do we order our elements when we talk about permutations? As noted before, this is for convenience.
+If we assign a natural order to the elements of $\Omega$ (say, 1234), we can identify permutations by simply listing
+their output:
+Clearly, $[1234]$ represents the trivial permutation, $[2134]$ represents \say{swap first two,}
+and $[4123]$ represents \say{cycle right.}
+
+\problem{}
+Draw string diagrams for $[4123]$ and $[2341]$.
+
+\vfill
+\pagebreak