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I am reading about cycle decompositions where a cycle $(a_1 a_2 a_3\cdots a_m)$ is defined as the permutation which sends $a_i$ to $a_{i+1}$, $1\leq i\leq m-1$ and sends $a_m$ to $a_1$. But I am unable to understand this particular notation: $(1 2 3)\circ ( 1 2)(3 4)$. What is this supposed to mean and how do we compute it? |
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You want to think of your permutations as functions. In your example, let $f:\{1,2,3,4\}\rightarrow\{1,2,3,4\}$ be the function represented by $(1,2,3)$. This means that \begin{align*} f(1)=2,\quad f(2)=3,\quad f(3)=1,\quad \text{and} \quad f(4)=4. \end{align*} In a similar way, let $g:\{1,2,3,4\}\rightarrow\{1,2,3,4\}$ be the function represented by $(1,2)(3,4)$. So \begin{align*} g(1)=2,\quad g(2)=1,\quad g(3)=4\quad\text{and}\quad g(4)=3. \end{align*} Then if we do $(1,2,3)$ and then do $(1,2)(3,4)$, then this is the same as doing $f$ and then $g$. So we get that \begin{align*} 1\stackrel{f}{\longmapsto}2\stackrel{g}{\longmapsto}1\\ 2\stackrel{f}{\longmapsto}3\stackrel{g}{\longmapsto}4\\ 3\stackrel{f}{\longmapsto}1\stackrel{g}{\longmapsto}2\\ 4\stackrel{f}{\longmapsto}4\stackrel{g}{\longmapsto}3.\\ \end{align*} So we may represent the effect of doing $f$ and then doing $g$ by $(2,4,3)$. Since this is the effect of doing $(1,2,3)$ and then doing $(1,2)(3,4)$, you have that \begin{align*} (1,2)(3,4)(1,2,3)=(2,4,3) \end{align*} if you are composing permutations from right to left. Note: If you are composing permutations from left to right then you would get \begin{align*} (1,2)(3,4)(1,2,3)=(1,3,4). \end{align*} |
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It means composition of permutations - you first do one and then the other. Whether you do the left-hand permutation first or the right-hand one depends on the convention you're using, and both sometimes occur. To compute it, you should see what it does to each element $1,2,3,4$. To give an example, let's assume that you first perform the permutation $(12)(34)$, and then $(123)$ (you should check if this is the convention you are using). Then to see where the composition sends the element $1$, we apply the first permutation, which sends it to $2$. Then we apply the second permutation, which sends this $2$ to $3$. So the composition maps $1$ to $3$, and in cycle notation will have a cycle beginning $(13\cdots)$. |
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