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Added cycle notation

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Mark
2023-12-18 10:55:08 -08:00
parent 0319c52248
commit bd88b894a1
3 changed files with 423 additions and 25 deletions
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49
Advanced/Symmetric Group/parts/0 intro.tex
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@ -44,7 +44,7 @@ Why do we define permutations as a \textit{bijective} map?
We can visualize permutations with a diagram we'll call the \say{braid.}
The arrows in the diagram denote the image of $f$ for each possible input.
The arrows in this diagram denote the image of $f$ for each possible input.
Two examples are below:
\vspace{2mm}
@ -60,10 +60,10 @@ Two examples are below:
\node (4b) at (2, -2) {4};
\node (2b) at (3, -2) {2};
\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
\line{1a}{1b}
\line{2a}{2b}
\line{3a}{3b}
\line{4a}{4b}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[scale=0.5]
@ -77,12 +77,14 @@ Two examples are below:
\node (3b) at (2, -2) {3};
\node (4b) at (3, -2) {4};
\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
\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),
@ -113,10 +115,10 @@ The rightmost diagram uses arbitrary, meaningless labels.
\node (3b) at (2, -2) {3};
\node (4b) at (3, -2) {4};
\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
\line{1a}{1b}
\line{2a}{2b}
\line{3a}{3b}
\line{4a}{4b}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[scale=0.5]
@ -130,10 +132,10 @@ The rightmost diagram uses arbitrary, meaningless labels.
\node (3b) at (2, -2) {3};
\node (2b) at (3, -2) {2};
\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
\line{1a}{1b}
\line{2a}{2b}
\line{3a}{3b}
\line{4a}{4b}
\end{tikzpicture}
\hfill
\begin{tikzpicture}[scale=0.5]
@ -147,12 +149,14 @@ The rightmost diagram uses arbitrary, meaningless labels.
\node (3b) at (2, -2) {$\circledcirc$};
\node (4b) at (3, -2) {$\boxdot$};
\draw[line width = 0.3mm, ->, ocyan] (1a.south) -- (1a.south |- 0, -0.5) -- (1b.north |- 0,-1) -- (1b.north);
\draw[line width = 0.3mm, ->, ocyan] (2a.south) -- (2a.south |- 0, -0.5) -- (2b.north |- 0,-1) -- (2b.north);
\draw[line width = 0.3mm, ->, ocyan] (3a.south) -- (3a.south |- 0, -0.5) -- (3b.north |- 0,-1) -- (3b.north);
\draw[line width = 0.3mm, ->, ocyan] (4a.south) -- (4a.south |- 0, -0.5) -- (4b.north |- 0,-1) -- (4b.north);
\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:
@ -167,7 +171,6 @@ the same permutation is depicted in all three diagrams. This example demonstrate
\vspace{2mm}
\vspace{1cm}
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
@ -182,7 +185,7 @@ Draw braids for $[4123]$ and $[2341]$.
Finally, note that permutations (as defined in \ref{permadef}) are \textit{not} \say{orderings of a certain set.} \par
They are defined as \textit{bijective maps}, which can be \textit{thought of} as orderings. \par
They are defined as \textit{bijective maps}, which can be written as orderings of a given array. \par
Remember: permutations are verbs!
\pagebreak