Cycle Decomposition/Examples
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Examples of Cycle Decompositions
Permutation in $S_7$
Consider the permutation given in two-row notation as:
- $\rho = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 6 & 1 & 7 & 3 & 2 & 4 \end{pmatrix}$
The cycle decomposition for $\rho$ is:
- $\begin{pmatrix} 1 & 5 & 3 \end{pmatrix} \begin{pmatrix} 2 & 6 \end{pmatrix} \begin{pmatrix} 4 & 7 \end{pmatrix}$
Permutation in $S_9$
Consider the permutation given in two-row notation as:
- $\rho = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 1 & 4 & 6 & 2 & 8 & 9 & 7 & 5 & 3 \end{pmatrix}$
The cycle decomposition for $\rho$ is:
- $\begin{pmatrix} 1 \end{pmatrix} \begin{pmatrix} 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 5 & 8 \end{pmatrix} \begin{pmatrix} 7 \end{pmatrix}$
or, omitting the $1$-cycles:
- $\begin{pmatrix} 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} 5 & 8 \end{pmatrix}$
Product of Permutations in $S_9$
Consider the product permutations:
- $\rho = \begin{pmatrix} 1 & 2 & 4 \end{pmatrix} \begin{pmatrix} 3 & 5 & 7 & 9 \end{pmatrix} \begin{pmatrix} 1 & 3 & 9 \end{pmatrix} \begin{pmatrix} 2 & 3 & 4 & 5 & 6 & 8 \end{pmatrix}$
Then $\rho$ evaluates to:
- $\begin{pmatrix} 1 & 5 & 6 & 8 & 4 & 7 & 9 & 2 & 3 \end{pmatrix}$